www.elsevier.com/locate/earscirev

Earth-Science Reviews 63 (2003) 295–349

Modeling shear zones in geological and planetary sciences:

solid- and fluid-thermal–mechanical approaches

K. Regenauer-Lieba,*, D.A. Yuenb,1

a Institute of Geophysics, ETH Zurich, Honggerberg HPPO15-Gebaude, CH-8093 Zurich, SwitzerlandbDepartment of Geology and Geophysics, Supercomputer Institute, University of Minnesota, Minneapolis, USA

Received 15 April 2002; accepted 4 March 2003

Abstract

Shear zones are the most ubiquitous features observed in planetary surfaces. They appear as a jagged network of faults at the

observable brittle surface of planets and, in geological exposures of deeper rocks, they turn into smoothly braided networks of

localized shear displacement leaving centimeter wide bands of ‘‘mylonitized’’, reduced grain sizes behind. The overall size of

the entire shear network rarely exceeds kilometer scale at depth. Although mylonitic shear zones are only visible to the observer,

when uplifted and exposed at the surface, they govern the mechanical behavior of the strongest part of the lithosphere below

10–15 km depth. Mylonitic shear zones dissect plates, thus allowing plate tectonics to develop on the Earth. We review the

basic multiscale physics underlying mylonitic, ductile shear zone nucleation, growth and longevity and show that grain size

reduction is a symptomatic cause but not necessarily the main reason for localization. We also discuss a framework for analytic

and numerical modeling including the effects of thermal–mechanical couplings, thermal-elasticity, the influence of water and

void-volatile feedback. The physics of ductile shear zones relies on feedback processes that turn a macroscopically

homogenously deforming body into a heterogeneously slipping solid medium. Positive feedback can amplify strength

heterogeneities by cascading through different scales. We define basic, intrinsic length scales of strength heterogeneity such as

those associated with plasticity, grain size, fluid-inclusion and thermal diffusion length scale.

For an understanding ductile shear zones we need to consider the energetics of deformation. Shear heating introduces a

jerky flow phenomenon potentially accompanied by ductile earthquakes. Additional focusing due to grain size reduction

only operates for a narrow parameter range of cooling rates. For the long time scale, deformational energy stored inside the

shear zone through plastic dilation or crystallographic- and shape-preferred orientation consumes only a maximum of 10%

of energy dissipated in the shear zones but creates structural anisotropy. Shear zones become long-living features with a

long-term memory.

A special role is attributed to the presence of water in nominally anhydrous minerals. We show that water directly

affects the mechanical equation of state and has the potential to synchronize viscous and plastic flow processes at

geological time scale. We have shown that fully coupled finite element calculations, using mechanical data from the

laboratory, can reproduce the basic mode of deformation of an entire mylonitic shear zone. The next step of modeling lies

in benchmarking basic feedback mechanism in field studies and zooming into the braided network of shear zone structure,

0012-8252/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0012-8252(03)00038-2

* Corresponding author. Present address: CSIRO Exploration and Mining, 26 Dick Perry Ave., Perth, WA 6151, Australia.

Fax: +61-864368555.

E-mail addresses: klaus.regenauer-lieb@csiro.au (K. Regenauer-Lieb), davey@krissy.msi.umn.edu (D.A. Yuen).1 Fax: +1-612-6253819.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349296

without losing the large-scale constraint. Numerical methods capable of fulfilling the goal are emerging. These are adaptive

wavelet techniques, and hybrid particle–finite element codes, which can be run over a computational GRID across the net.

D 2003 Elsevier Science B.V. All rights reserved.

Keywords: plate tectonics; shear localization; mylonitic shear zones; energetics; multiscale modeling

1. Introduction

Shear zones in geology occur over many different

length scales, from micro (grain size)- to large plate

boundary scale (Fig. 1). Plate boundaries define plate

tectonics. Shear zones are found on the Earth, Venus,

Mars and icy planets such as the Jovian, and Saturnian

Moons (Fig. 1). The San Andreas fault zone is an

Fig. 1. Shear zones on (a) Venus: Guinevera Planita showing equidistant wri

Europa: ice ridges and grooves forming a criss-cross structure on the Jovian m

scales 1780� 1780 km. (c) Earth: The San Andreas Fault on the Earth (http:

mylonitc shear zone: (image scales 4� 4 cm), http://www.courses.eas.ualb

excellent example of a terrestrial large-scale brittle

fault zone (Luyendyk and Hornaflus, 1987; Luyendyk

et al., 1985; Lyzenga et al., 1986). Shear zone exam-

ples with plate scale ductile flow localization are the

Alpine Fault in New Zealand (Wellman, 1984), the

Kun-Lun and the Altyn-Tagh shear zones in China

(Tapponnier and Molnar, 1977). Brittle fault zones can

be well traced at depth by narrow seismo-active linea-

nkles covering large parts of the surface; image scales 40� 40 km. (b)

oon (http://www.jpl.nasa.gov/galileo/europa/e4images.html); image

//www.scecdc.scec.org/faultmap.html). (d) Microstructural image of a

erta.ca/eas421/images/photographs/09 9-06qzmylonitegyp.jpg.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 297

ments. Active brittle fault zones can be analysed in

terms of their complexity and modes of dynamic

interaction by estimates of the fractal dimensions

(Matsumoto et al., 1992; Okubo and Aki, 1987), the

earthquake statistics (Wyss and Wiemer, 2000) and

detailed analyses of the rupture process (Sieh et al.,

1993; Zhao and Kanamori, 1993). Ductile fault zones

are only accessible when they are exposed to the

surface at which point they record a snapshot of the

geological history. Indirect observations on active

shearing are only available for extreme cases with

anomalous heat transfer (Hochstein and Regenauer-

Lieb, 1998; Hochstein et al., 1993) and again through

secondary observations of the earthquake rupture pro-

cesses (Wiens and Snider, 2001).

Insight into the physics, dynamics and mechanics

of ductile shear zones is, however, imperative for

understanding plate tectonics, because plate bound-

aries are defined by ductile shear zones (Bercovici,

1996, 2002; DeMets et al., 1990). Many approaches

have been developed to describe shear zones. The

important aspect of time scale has been emphasized

in the different fields using various rheologies such as

purely viscous on the long time scale to visco-elastic

on short time scales. Geodynamic modeling of ter-

restrial planets uses time scales as defined by cyclic,

quasi-periodic behaviour. On the Earth, this is known

as the Wilson cycle, on Venus the resurfacing time

scale, icy planetary surfaces have a comparatively

fast cycle of several hundred years. Earthquake

modeling goes down to a shorter time scale with

recurrence periods of earthquakes, which are less

than 10 ka and the process of the earthquake rupture

down to tens of seconds or a few minutes. Most

modeling approaches have not considered the cou-

pling of loading rate, the mechanics and energetics of

shear zones.

Laboratory analogue models (e.g. Faccenna et al.,

1996; Shemenda and Grocholsky, 1994) are obvious-

ly limited by availability of materials and laboratory

conditions. They fail to reproduce appropriate time

scales for the analysis of the delicate influence of

coupling thermal diffusion to dynamic gravity loading

rate as imposed by thermal expansion and to the shear

zone internal temperature sensitive properties. Purely

mechanical numerical analyses have been done, e.g.

by Buck, Poliakov and Pollitz (visco-elastic but

without energy) (Buck and Poliakov, 1998; Poliakov

et al., 1994; Pollitz, 2001). We are emphasizing in this

article on the nonlinear feedback by considering both

thermal–mechanical coupling (the energetics) and

using composite rheology, ranging from the simple

viscous to complex visco-elasto-plastic rheologies. In

doing so, we restrict our description to the depth

range deeper than 10 km, because the near surface

layers provide additional complexities without con-

tributing much to the strength of the lithosphere. We

will highlight the differences between the models and

point out where we need to consider dynamical time

scales arising in these thermal–mechanical systems.

In the sections to come we will describe three kinds of

models:

(1) Geodynamic modeling: time scale < 500 Ma,

(2) Earthquake modeling: time scale < 5 ka,

(3) Structural geological modeling no dynamical time

scale, only driven by boundary conditions.

Shear zones form as the result of a thermo-mechan-

ical instability that can have many different origins.

Prior to the formation of shear zones strain hardening

decreases to a critical level which depends on the

material, its current state and its p–T condition. The

scientific challenges to understanding the dynamics of

shear zones are the (sometimes) unobservable dynam-

ics and multiscale physics summarized in Tables 1 and

2. Solutions to the problem of brittle shear zone

formation will provide insight into the quasi-periodic-

ity of earthquakes while solutions to the ductile shear

zones gives insights into the problem of cyclic-like

nature of plate tectonics. This paper focuses on the

second problem.

Modern numerical approaches for understanding

the dynamics of brittle earthquakes are also discussed.

However, we will not go into the details of modelling

fault zones in the brittle domain because the multiscale

thermal–dynamic material properties are less well

constrained than the ductile properties. The brittle

strength of the lithosphere is probably overstated.

Significant scale dependence of the brittle properties

of rocks have been reported in the literature on the

brittle field (see e.g. Shimada, 1993). The brittle

compressive failure strength of a rock is for instance

one order of magnitude smaller at meter scale than at

cm scale. Above one meter there appears to be a

statistical satisfactory number of planes of weaknesses

Table 1

Dynamics and multiscale physics in brittle shear zones. Scientific challenges

Spatial scale Physics Input from

lower scale

Output to

upper scale

Computational

methods

Research status

Griffith crack

0.1–1 AmAtomic bonds Equilibrium lattice

spacing

Effective (damaged)

elasticity

Molecular dynamics,

finite elements (FE),

finite difference (FD)

Understand mechanics

and kinetics

Grain size

1 Am–1 cm

Atomic bonds/contact

interactions

Cohesive potential

across grains

Effective viscosity Particle dynamics Link gouge mechanics

to fault behaviour

Fault groups

100 m–10 km

Coarse grain, planar

fault, effective friction

Effective or rate and

state variable friction

Effective l,viscosity and

elasticity

Finite elements,

boundary elements

Understand dynamical

modes of faults

Tectonic plate

boundary

Earthquake dynamics Effective

visccoelasticity

No larger scale Finite elements Coupling to ductile

shear zones

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349298

in rocks so that the failure strength does not decrease

further. Unfortunately, very huge testing machines are

necessary to obtain mechanical data relevant for the

larger scale. The necessity for assessing the large-scale

has been realized only for the laboratory assessment of

friction (Dieterich, 1979a). An equivalent approach is

lacking for the compressive failure strength of rocks.

For ductile materials a scale-dependent strength

transition is well described between the micro-

(microns) and nanoscale (nm). It defines the intrinsic

Table 2

Dynamics and multiscale physics in ductile shear zones. Scientific challe

Spatial scale Physics Input from

lower scale

Point defects 1–5 A Diffusion by random

walk

Lattice vibrations

(phonons)

Burger vector of line

defects 5–10 A

Dislocation glide + climb Lattice vibrations

obstacles

Intrinsic length scale

of plasticity

0.2–2 Am

Geometrically necessary

dislocations

Burger vector,

shear modulus

Grain boundaries

1 Am–1 cm

2-D lattice defect

high + low angle

Lattice vibrations,

dynamic

recrystallization

Heat conduction

1 cm–100 km

(diffusivity/strain rate)0.5 Shear heating

Tectonic plate

boundary

Defines plate rotations Composite fault

rheology

length scale of plasticity (Gao et al., 1999). This scale is

more easily accessible to material testing. By analogy

to these ductile strain-gradient methods listed in Table

2, we appear to lack in Table 1 a theory that describes

the dynamical behavior of brittle material between

grain size and meter scale.

Outlining various sections to come, we will in

Section 2 summarize the basic equations underlying

shear zone formation. In Section 3, we will discuss

basic feedback in a one-dimensional shear cell. In

nges

Output to

upper scale

Computational

methods

Research status

Pipe diffusion,

lattice diffusion

Molecular dynamics

FD/FE

Non-equilibrium

thermodynamics

Statistical

distribution

of dislocations

Molecular dynamics

FD/FE

Dislocation,

mechanics

Nonlinear flow

law

FE Strain gradient

plasticity

Linear flow law Particle dynamics,

finite element

Mechanical equation

of state

Thermal

weakening

Fully coupled

FD+FE

2-D thermomechanical

shear zones

No larger scale Finite elements Self-consistent plate

tectonics

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 299

Section 4, we will expand the analysis into a two-

dimensional shear zone model and discuss the role of

fundamental length scales. These theoretical models

are applied in Section 5 to the problem of self-

consistent plate tectonics. Section 6 looks into the

problem of longevity and memory of shear zones,

while Section 7 points out the implications for possible

directions in the future of earthquake modeling. We

conclude in the summary with a synopsis of length

scales obtained from structural field observations and

highlight the implications for thermal–mechanical

processes inside the shear zone.

2. Mathematical equations

In the following, we will present relations between

the stress in a body and its associated cumulative strain

(solid mechanics) or strain rate (fluid mechanics). For a

comprehensive analysis of shear zones, we cannot

restrict ourselves with a one-dimensional analysis as

shown in Fig. 2, but we need to go to a full three-

dimensional formulation. We first define the basic

Fig. 2. Average macroscopic shear stress– strain diagram as recorded by

leading to the development of a shear zone after critical strain hardening

negative for the onset of a shear localization but can be positive for a varie

shear zone has reached its maximum width. The 1D shear zone shows the m

elasto-visco-plastic rheology.

quantities. The stress matrix describes the traction

being carried per unit area by any internal surface in

the body under consideration. This is the ‘‘Cauchy

stress’’, which is given by

Cauchy stress riju

r11 r12 r13

r21 r22 r23

r31 r32 r33

0BBBB@

1CCCCA ð1Þ

where rij is the force per area acting on surfaces facingin the i-direction and pulling/pushing it in the j-direc-

tion. We will always imply the so-called ‘‘Einstein

convention’’ (Hill, 1950), i.e. a summation over re-

peated indices. The Cauchy stress gives the ‘‘true’’

stress for any particular choice of orientation of the

coordinate system. It is useful to derive independent

invariants from the Cauchy stress:

Pressure p ¼ 1

3rkkdij� �

¼ 1

3r1 þ r2 þ r3ð Þ ð2Þ

This is also known as the trace, first invariant or

isotropic part of the stress tensor. dij is the Kronecker

the gauges of a laboratory experiment showing typical conditions

for different strain rates. Strain hardening must not at all cost be

ty of materials, if the confining pressure is low. At steady state, the

athematical idealization with viscous, visco-elastic, elasto-plastic or

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349300

delta, which is one when the indices are equal and

zero for unequal indices. We will use the term

‘‘pressure’’ p. Most flow laws are independent of

pressure so that it is convenient to define the flow

law on the basis of the square root of the second

invariant of the deviatoric stress tensor which itself is

defined by subtracting the pressure from the Cauchy

stress rijV= rij� p.

Effective Stress rV¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2rijVrijV

rð3Þ

This invariant of the deviatoric stress tensor is also

known as the ‘‘root mean square’’, or the ‘‘effective

shear’’ stress, which is a scalar. Norms differing by a

factor can be found in the literature (Chakrabarty,

2000; Ranalli, 1995). In Appendix A, we discuss and

Fig. 3. The second invariant of the deviatoric stress tensor traces a cylinde

space (principal stresses are the axes of a 3D Cartesian space). In the ideal

(Chakrabarty, 2000). The inside of the cylinder gives an elastic stress state

while any plastic strain is possible when the cylinder is reached. In the gen

Eq. (12)) or contract (strain soften to be discussed later) as a function of th

must be conjugate as indicated by the vectors in the deviatoric stress space

the strain increment vector must always be normal to the yield surface (p

cylinder is lacking (except for the Bingham solid) but conjugacy between

system, the incremental principal strain, must be replaced in the viscous c

explain the alternative definition of the effective

stress, which is more convenient for implementing

experimental flow laws into numerical models. In

Appendix B, we show how to turn the scalar effec-

tive stress–strain rate relation into tensorial flow

laws.

For the choice of definition of strain and strain rate,

it is necessary to consider energy dissipated by the

deformation (Fig. 3), i.e. we want to define a strain that

when multiplying its increment deij with the Cauchy

stress gives the work done in the unit body. This

collapses to a one-dimensional case to the familiar

definition of work by force times displacement. In

Fig. 3, we show the case of ‘‘associated plasticity’’,

i.e. co-axial stress and strain increment tensors (Ap-

pendix B). This is a necessary condition when plastic

deformation does not imply volume (surface energy) or

r around the first invariant p when visualized in the principal stress

rigid-plastic case, the cylinder defines the von Mises Yield Envelope

(in this section E =l, i.e. rigid, we will relax this assumption later),

eral plastic case, the cylinder is allowed to swell (strain harden, see

e plastic strain. In the absence of dilatancy, stress and strain tensors

(white circle going through the origin also known as the p-plane), i.e.lastic normality rule). For the case of a viscous flow rule, the initial

stress and strain rate still holds. The second superposed coordinate

ase by the principal strain rates.

Fig. 4. Bingham (Nye, 1953), Ludwik’s dynamic plasticity law

(Ludwik, 1909) and the Peierl’s stress flow law (Ashby and Verall,

1977). The Bingham and the two exponential laws have a yield

phenomenon; however, the latter two are highly non-linear after

yield. The Bingham solid also lacks the aspect of starting to creep

with a characteristic strain rate after yield.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 301

other changes of energy. The infinitesimal strain incre-

ment tensor for time dt is

Strain increment and strain rate

deij ¼1

2

BdXi

Bxjþ BdXj

Bxi

� �; eij ¼

deijdt

ð4Þ

where dXi is the infinitesimal displacement of a particle

in time dtwith a current position vector xi. Note that, in

linear elasticity, it is customary to omit the increment

and use the above definition as a small strain measure

eij. In plasticity, an appropriate integration is mandatory

unless proportional straining is assumed. Analogous to

the definition of invariants for stress we define the roots

of the second strain increment and strain rate invariants

as ‘‘effective’’ strain increment or strain rate:

Effective strain and strain rate

de ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2deijdeij

r; e ¼

ffiffiffiffiffiffiffiffiffiffiffiffi1

2eijeij

r: ð5Þ

2.1. No elasticity and compressibility

Consider slow motion. We neglect inertial forces

and describe a balance of all forces in a unit volume by

Momentum conservationdrij

dxjþ Bi ¼ 0 ð6Þ

whereBi is the sum of the body forces.We introduce the

plastic flow rule of a solid and the viscous flow rule of a

fluid and show their relation. In plasticity or fluid dy-

namics, the strain increment or strain rate, respectively,

are given by a function of the effective shear stress

Plastic and Viscous Flow Rule de ¼ f ðrVÞ; e ¼ f ðrVÞð7Þ

However, plasticity deals with the stress-strain

relation, while the strength of fluids is described by

the strain rate. We point out here that the plastic flow

rule is dimensionally consistent, i.e. time does not

enter in the equations but it certainly appears in the

viscous flow rule (Hill, 1950). Associated flow

implies that the strain increment or the strain rate is

everywhere normal to the flow potential (Fig. 3)

whether there exists a finite yield surface (plasticity)

or a continuous potential (viscous flow).

In the following nomenclature, we will separate

plasticity-based formulations as defined by the first

part of Eq. (7), usually describing a yield phenomenon

attributed to the dislocation controlled flow, from fluid

dynamic approaches, mostly characterized by diffusion

without a yield phenomenon, by the second part of Eq.

(7). Plasticity often involves the consideration of elas-

ticity while in fluid dynamics elasticity is frequently

neglected. We will, however, also discuss hybrid

rheologies that comprise elasticity and viscosity, or

elasticity and plasticity and present results for the

complete rheological elasto-visco-plastic approach.

Examples for visco-plastic flow rules are:

Bingham0s Viscous Flow Rule

e ¼ f ðrVÞ ¼e ¼ 0; rV<ry

e ¼ 12ge

ðrV�ryÞ; rVzry

ð8Þ

8><>:

where ge is the effective viscosity. The Bingham solid

incorporates plasticity into the standard Newtonian vis-

cous flow rule. Newtonian viscous flow without plas-

ticity is recovered when the yield stress ry = 0 (Fig. 4).

Power�law Viscous Flow Rule e ¼ f ðrVÞ¼a�nðrVÞn

ð9ÞFor modeling the lithosphere, this power-law is

often used with various values of exponent n (Chester,

1995; Christensen, 1992; Lenardic et al., 1995). It

varies between n= 1 for standard Newtonian viscous

flow (a = 2ge), over n = 3–4.5 from laboratory data,

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349302

and to a very high n (n= 35) to reflect a pseudoplastic

flow law (Fig. 5). For n!l, we recover the ideal

rigid-plastic flow law with a constant yield stress

rV= ry = a for any strain rate (Nye, 1953). A third

way of incorporating plasticity into viscous flow is

the exponential flow law.

Ludwik0s dynamic plasticity law

e ¼ e0exprV� r0

b

� �ð10Þ

solved for stress rV¼ r0 þ blnee0

� �; eze0 ð11Þ

where r0, b and e0 are material constants. The ‘‘over-

stress’’ formulation of this flow law is obvious, when

the flow law is solved for stress, giving the logarithmic

Eq. (11). Here, r0 is the initial yield stress associated

with a characteristic strain rate e0 for the onset of creep.For higher strain rates, the stress increases by an

increment that is controlled by the logarithmic term.

Up to now, we have discussed extensions of viscous

flow theory, thus allowing incorporation of ideal

plasticity into the viscous flow. In the classical New-

tonian viscous case viscosity is the only material

parameter introducing time dependence in the flow

rule. For incorporating plasticity, at least one addition-

al material parameter is necessary for scaling acurately

the yield stress.

Classical plasticity uses the same Bingham style

flow rule, but is independent of time. Assuming

Fig. 5. Power law showing the yield like phenomenon (pseudo-

plastic behaviour) for high n. The Newtonian viscous flow with

n= 1 separates a regime where viscosity increases with increasing

strain rate (n< 1), i.e. shear thickening flow, and the viscosity

decreases with increasing strain rate (n>1), i.e. shear thinning flow.

Shear thickening/thinning describe the tendency of fluids to develop

wider/narrower shear zones with increasing strain rates.

proportional straining, the integrated strains replace

the incremental strain and we can come up with an

analogous equation.

Rigid Plastic Flow Rule

e ¼ f ðrVÞ ¼e ¼ 0; rV< ry

e ¼ c�nðrVÞ�n; rVzry

8<: ð12Þ

This is the popular power-law hardening law

(Ludwik, 1909) that describes strain (work) hardening

by two material parameters the constant stress c and

dimensionless n. This particular form of strain hard-

ening can be derived from the theory of defects

(Hirsch, 1975). It describes the increasing strength

of a crystalline solid owing to an increase in disloca-

tion density as the strain increases. Again, if n!l,

we recover the ideal rigid-plastic body where the

plastic stress does not increase with strain.

In sum, we can now come up with an extension to

Ludwik’s visco-plastic formulation with strain hard-

ening/weakening,

Generalized dynamic plasticity law

rV¼ f ðeÞ þ bðeÞf ðeÞ ð13ÞNow, the yield stress f (e) and the stress scale factor

b(e) of the flow term are also a function of plastic

strain. Eqs. (1)– (13) provide a complete set for

solving the momentum-rheology equations.

2.2. Add temperature and pressure but without water

When considering temperature in addition, we

have to solve for the energy equation. Using a

Lagrangian framework, i.e. the equation is solved

with reference to a moving particle indicated here

by the substantial derivative DT/Dt in which case the

advection of heat drops out of the Eulerian equation.

In considering temperature as the thermodynamic

variable (Appendix C), we obtain

Temperature Equation

qCp

DT

Dt¼ vrVeij � jqCpj

2T þX

Xi ð14Þ

where j is the diffusivity, q the density, Cp the

specific heat, v the mechanical heat conversion effi-

ciency and Xi the sum of additional heat sources such

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 303

as radiogenic heating, heat of solution, phase transi-

tion, chemical reaction and other terms discussed in

the chapter on volatiles and equation of state.

In the theory of thermally activated creep, temper-

ature and pressure enter through an exponential ther-

modynamic term with Arrhenius dependence

Arrhenius term e ¼ f ðrVÞexp �Qþ pV

RT

� �ð15Þ

where Q is the activation energy of the particular flow

creep mechanism, V the corresponding activation

volume and R the universal gas constant. The Arrhe-

nius term in the exponential is additionally controlled

by the presence of water.

2.3. Add grain size

Up to now, we have given equations for creep

based on line defects, i.e. dislocation glide (the Peierls

stress mechanism, a Ludwik’s law style flow law) and

dislocation climb (power-law) processes. Plasticity by

line defects gives mixed plastic and viscous constitu-

tional properties, and hence must be described by a

dynamic plasticity law. When scaling down to point

defects, there is, however, one flow process that

deserves the label of fluid dynamic approaches. This

is the diffusional flow which relies on grain size

sensitive creep.

Grain size sensitive creep

e:D ¼ e0g0

g

� �m

rVsexp � QD þ pVD

RT

� �ð16Þ

where g0 is the initial grain size, g the current grain

size and m the grain size exponent (between 0.3 and

0.5) (Van Swygenhoven, 2002) and s the stress

exponent which is close to unity (Mei and Kohlstedt,

2000). Positive feedback comes in during the defor-

mation through strain-dependent grain size reduction

given by

Grain size reduction

dg

Bt¼ keP gr � gð Þ þ k0

gexp � H

RT

� �ð17Þ

where k, k0 and H are material constants. The first

term on the right-hand side describes the effect of

grain size reduction and the second term describes the

effect of grain growth (Karato, 1989).

2.4. Add water

Water changes the rheology in several ways. We

first discuss the effect when only minor quantities of

water are added to nominally anhydrous minerals, i.e.

the rock incorporates water into the solid without

microstructural modification. In this case water has

two major effects.

Water weakening e ¼ af ðrVÞexp � Q*þ pV*

RT

� �ð18Þ

Water changes the activation volume and the

activation energy in the Arrhenius term because of

the formation of new hydroxyl bonds. It accelerates

creep rates by a scalar factor a, which ranges between

0.1 and unity depending on water content (Jung and

Karato, 2001). The activation energy Q* is lower than

the dry value but only by about 10%. The activation

volume term V* that is attributed to water changes its

pressure sensitivity. Large variations up to 50% are

reported in the literature. Its value is difficult to

determine with precision, but the general magnitude

will give a sense of the physics.

Since water has the same principal effect on all

creep mechanisms, it is most prominent in highly non-

linear flow laws, especially where it has a rather

strong influence on degree of non-linearity. The

power-law has already been introduced as a non-linear

flow law. The pre-exponential weakening factor a

linearly scales the magnitude of the yield like transi-

tion from high viscosity at low strain rates to low

viscosity at high strain rates. However, the sharpness

of the transition is not affected since it is only

controlled by the exponent n. For high water content,

the overall weakening through the addition of water

can reach an order of magnitude. For rocks the power-

law, exponent never exceeds n = 5, so we cannot call

this a true yield phenomenon as in the pseudo-plastic

case. Flow laws, where water has a strong effect on

the yield stress, should indeed play a prominent role in

the nucleation of shear zones. Water indeed has a

fundamental influence on the exponential flow law

that has been applied to indentation hardness experi-

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349304

ments of quartz and olivine (Evans, 1984; Evans and

Goetze, 1979; Goetze and Evans, 1979). It is also

known as ‘‘low temperature plasticity’’, ‘‘Peierls

stress’’ or ‘‘Dorn-Harper’’ creep law.

Peierls strain� stress law

eL ¼ e0aexp � QL*þ pVL

*

RT1� rV

r0

� �2 !

ð19Þ

This law is more complex than Ludwik’s dynamic

plasticity law (Eq. (11)) in that it has also an addi-

tional sharp transition at high stress. We show here

that water has an influence on its first embedded yield

criterion. Inverting Eq. (19), we obtain

Peierls stress� strain law

rV¼ r0 � r0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� RT

QL*þ pVL

*

� �ln

eLe0a

� �s; eL > eL0

ð20Þ

This flow law recovers the ideal plastic case for a

hypothetical T= 0 K when the second term vanishes

and rV= r0 = ry is the ideal yield stress, the so-called

‘‘Peierls stress’’, for any strain rate. The reference

strain rate e0 is a material constant. The thermally

activated second term influences the yield phenome-

non and yielding occurs with a characteristic strain

rate e0L (Branlund et al., 2001; Regenauer-Lieb et al.,

2001).

Characteristic strain rate after yield

e:L0 ¼ ae0exp � QL*þ pVL*

RT

� �ð21Þ

In analogy to the discussion of Ludwik’s style

dynamic plasticity law (Eq. (11)), we now obtain a

characteristic strain rate (Fig. 4) for the onset of

creep that is no longer a material constant but

depends on pressure, temperature and water content.

Consequently, the initial yield stress also depends on

these thermodynamic quantities and the water con-

tent.

We have thus far given given constitutive equa-

tions for dynamic visco-plasticity for the case of low

water content. If volatiles in excess of their solubility

are present in the solid, volatiles will act as a damag-

ing agent, i.e. they will create voids (fluid inclusions),

which weaken the rock matrix as considered by the

strain-dependent parameter in the generalized dynam-

ic plasticity formulation. In Section 2.5, we will

describe a self-consistent strain-weakening, theory

arising from influx of volatiles.

2.5. Add volatiles

The von Mises yield envelope shown in Fig. 3 is

insensitive to the pressure. The lithostatic pressure

ensures that the mechanically strong part of the

lithosphere is in an overall compressive regime.

Therefore, the von Mises envelope is a safe approach

for large overburden pressure and for the case of

absence of volatiles. However, when volatiles are

present, the fluid pressure compensates the overbur-

den around the inclusion. For simplicity, we will

assume that the fluid pressure is lithostatic, if there is

no other load than a pure lithostatic load. This

implies that tensile stress states are possible locally

around the fluid under the addition of a tectonic load

(Petrini and Podladchikov, 2000).

When dealing with this problem mathematically

we have to bear in mind that the volatile content of a

deep seated rock is not more than 0.5 wt.% equiv-

alent to 3% of volume of the intact rock. This is the

largest content of fluid inclusions reported in the

literature (Roeder, 1965). We therefore cannot imply

a macroscopic brittle failure criterion on the basis of

a global effective stress principle. Instead, we have

to define the local void volume as a fracture con-

trolling parameter. If we define the relative density of

the rock by r, which is the ratio of solid over the

total volume, then we obtain the void volume

fraction as

Void Volume Fraction f ¼ 1� r ð22Þ

Because of the small void volume fraction, we can

only assume at a depth, say greater than 10 km, that

the bulk of the rock matrix is still deforming by

crystalline plasticity. The yield criterion therefore is

still based on the von Mises Criterion with an exten-

sion for pressure sensitivity and void volume fraction.

A hyperbolic cosine surface has been suggested

(Tvergaard, 1987) that truncates the von Mises style

Fig. 6. Yield envelope modified for the presence of volatiles. Volatiles limit the yield envelope in the tensile domain. During deformation the

void volume ratio fi describing the damage caused by volatile sheets, develops dynamically. Here a case is shown where f2>f1 and the yield

envelope is considerably weaker after damage. In the compressive domain, it approaches the von Mises yield surface shown in Fig. 3.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 305

envelope in the tensile domain as a function of the

void volume fraction.

Tvergaard Yield Criterion

rVry

� �2

þ2q1 f cosh q2ffiffiffi3

p p

ryV

� ���1þ q3 f

2

�¼ 0

ð23Þ

where the first term is the classical von Mises yield

envelope, the second term is important for tensile stress

states and the third term tracks the damage created by

the voids, q1,2,3 are material parameters. The yield

envelope is shown in Fig. 6. The dynamic evolution

of volatiles is given by the sum of the nucleation rate of

new voids plus the growth rate of existing voids

denoted by subscripts

Dynamic void evolution f ¼ fnuc þ fgr ð24Þ

where the growth rate is controlled by the mass

conservation

Void growth rate fgr ¼ eð1� f Þ ð25Þ

and the nucleation is assumed to be either strain rate

controlled

Ductile void nucleation rate fnuc ¼ Aeh ð26Þ

or stress rate controlled

Brittle void nucleation rate fnuc ¼ BðrVþ pÞ ð27Þ

where A and B are normal distributions around the

nucleation stress or strain for plastic strain (ductile) or

plastic stress (brittle) controlled nucleation (Needle-

man, 1994) and eh describes the corresponding hard-

ening strain rates. Both Eqs. (26) and (27) describe

material specific energy density rates for nucleation of

voids. In the ductile case, voids are nucleated due to

dislocation climb and glide while in the brittle theory

they are nucleated by classical cleavage cracks. Expres-

sions for accelerated growth after reaching a critical

void volume fraction can be found in the cited literature

(Dodd and Baiy, 1987).

We note here that the last three equations also feed

back into the energy Eq. (14) through the shear heating

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349306

efficiency v. The generation of new surface energy

stores some fraction of the deformational work to be

converted into heat later, thus lowering v. However, thelow void volume ratio considered here, implies that

other structural defects such as energy absorbed into

increasing the dislocation density are dominant. There-

fore, in solid mechanics, it is common practice to obtain

v from experiments such as thermographic imaging

techniques (Chrysochoos and Belmahjoub, 1992) and

consider the stored energy fraction 1� v by the pre-

factor v (see also Appendix C). For large strain, the

shear heating efficiency v of most materials is found to

lie between 85% and 95%. This means that almost all of

the deformational work is converted into heat and very

little is stored in microstructural processes.

A very similar fluid mechanical approach in two-

phase systems has been developed recently (Berco-

vici and Ricard, 2003; Bercovici et al., 2001a,b;

Ricard et al., 2001), which has the advantage of

considering fully the thermodynamic work including

explicit terms for void volume fraction (Eqs. (65)

and (66), Bercovici et al., 2001a). Being a viscous

approach, it can only describe long timescale pro-

cesses. This approach ignores, however, the duality

in the physics of void creation as it is expressed in

Eqs. (26) and (27). At present, the theory of Berco-

vici and Ricard is tuned to the brittle top 10 km

where damage is accommodated by brittle micro-

cracking.

2.6. Add equation of state and compressibility

The constitutional laws laid out above have been

formulated for a rigid-plastic, incompressible viscous

medium only defining its deviatoric strength. Al-

though the equation of state has already been used

implicitly in the flow laws, an important element of

physics is missing. Incompressible media are mathe-

matical idealizations, not to be found in nature. We

start here with the ideal gas, which has no infinite

compressible strength, its density being described by

the equation of state. Avogadro showed that under the

same p–T conditions the number of molecules in a

given volume is constant.

Ideal Gas Equation of State p ¼ n

VRT ¼ qmolRT

ð28Þ

where n is the number of mol in the volumeV (we recall

1 mol is defined on the basis of the quantity of carbon

isotopes contained in 12 g of 12C which is 6.022136�1023 mol� 1). The ideal gas constant R scales the

proportionality between thermal pressure and internal

energy. Defining the molar density qmol as the inverse

of the molar volume, the equation of state of water is

formulated (Pitzer and Sterner, 1994). Water has no

deviatoric strength and the non-ideal Helmholtz free

energy (see Appendix C) with the addition of the ideal

gas term of water is of following type

Water equation of state

p ¼ RTðqmol þ ciðTÞq2mol þ

XciðTÞHiÞ ð29Þ

where Hi contains higher orders of qmol and tempera-

ture sensitive coefficients ci(T) as well as two exponen-

tial terms necessary near critical point of water. For a

general solid with less extreme property changes,

similar expressions can be found (Dorogokupets,

2001; Poirier and Tarantola, 1998).

We are now in a position to reconsider the ener-

getics of compressibility. In the energy Eq. (14), we

have only looked at work done under deviatoric stress.

Compressibility introduces volume changes which are

recoverable upon unloading. An additional recover-

able term related to the elastic volume change appears

in the energy equation (see Appendix C).

qCp

DT

Dt¼ vrijV eij þ kthTequ

Dp

Dt� qCpjj

2T þX

Xi

ð30Þ

where kth is the thermal expansivity which multiplied

by the adiabatic temperature change Tequ describes the

recoverable elastic volume change. The energy equa-

tion and the equation of state are coupled equations

for pressure and internal energy (Appendix C). Si-

multaneously solving for the equation of state, the

rheology and the energy equations is a fundamental

prerequisite to understanding the complete thermo-

mechanical structure of shear zones.

2.7. Additive strain rate approximation

We have separated out the energetics of deforma-

tion into a conservative and dissipative component.

Elastic work is stored reversibly as strain energy while

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 307

plastic/viscous work is either dissipated immediately

as heat through viscous processes, or stored as struc-

tural defects. In addition, there are important differ-

ences in the intrinsic length and time scales of elastic

and plastic and viscous deformation. Visco-plastic

deformation is transmitted by the motion of line

defects, so there is a rate limited process controlled

by atomic relaxation times. Elastic strain relies on

electromagnetic waves, so it is determined by elec-

tronic relaxation times. The length scale of plastic

deformation relies on the size of line defects and

magnitude of lattice vibrations. The length scale of

viscous deformation relies on thermal and chemical

diffusion through lattice and crystal sizes, while

elastic deformation only relies on electronic (ionic

or covalent) potential in a perturbative sense. Since

theses processes are fundamentally different and op-

erate separately, a single elasto-visco-plastic constitu-

tional equation does not exist. Instead, we use the

additive strain rate decomposition.

Additive strain rate decomposition

eij ¼ eEij þ eTij þ eLij þ ePij þ eDij ð31Þ

where the total strain rate is a composite of elastic,

thermal (appearing through the recoverable thermal

expansion), Peierls, power-law and diffusion creep

strain rates are identified by their superscripts. Also

inherent in this assumption is that the deviatoric

strength, not defined in the equation of state, can be

added on the basis of the deviatoric properties of

creep, plasticity and elasticity laws. This composite

rheological law is the current continuum mechanics

approach to thermodynamics in the deviatoric stress

space. Recent non-equilibrium molecular dynamics

calculations have lent strong support (Holian and

Lomdahl, 1998) to these macroscopic simplifica-

tions.

2.8. Localization mechanisms

2.8.1. Constitutive theory

In geodynamics, it is common practice to consider

temperature and pressure only through their immedi-

ate effect on the strength parameters, without consid-

ering a fully coupled feedback (see Yuen et al.,

2000). We will refer to these approaches as mechan-

ical models. More elaborate approaches consider heat

conduction (second term in Eq. (14)) and radioactive

heat generation. Then, it suffices to calculate a

thermal solution say every several million years

because of the slow pace of conduction and radioac-

tive heat generation. Such approaches are sometimes

called thermo-mechanical (Beaumont et al., 1996b).

For the topic discussed here, this is a misnomer. A

staggered momentum and thermal solution does not

have the potential for thermal feedback. Hence,

staggered solutions must nucleate shear zones the

same way the mechanical models do. For the purpose

of nucleation shear zones, they also belong to the

mechanical group implying that flow localization is

entirely characterized by temperature sensitive-con-

stitutive laws.

The mechanical way of solving the problem is

basically thermally decoupled and it is necessary to

nucleate shear zones through the feedback between

momentum and rheology only (Poliakov and Herr-

mann, 1994; Tommasi et al., 1995). It is amenable to

full theoretical assessment if the rate effects are also

neglected (Rice, 1977). In this approach shear zone,

formation is understood as an instability that can be

predicted from the pre-localization constitutive equa-

tions.

Conditions are sought at which some small

perturbation is allowed accelerated growth so that

initial uniform smoothly varying deformation turns

into highly localized deformation, a flow bifurcation

occurs. Shear zones form on potential shear planes

if the strain hardening on those potential planes is

lower than a critical strain hardening hcrit depending

on rheology (Fig. 2). While the actual hardening is a

function of the deformational history, it is possible

to predict the tendency to spontaneous localization

by the magnitude of the hardening parameter for

specific rheologies, which turns out to be a material

parameter.

In earlier papers on localization (see Needleman

and Tvergaard, 1992; Rice, 1977 for reviews), much

work has been devoted to theoretical ends for

understanding constitutive instability (Appendix B).

The Earth’s brittle crust localizes readily in the form

of brittle shear zones, being entirely described by the

constitutive theory. However, according to the same

theory, the ductile part of the lithosphere cannot

localize. In mechanical models, the ductile part of

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349308

the lithosphere can only be deformed by way of

homogenous shear. Mechanical models therefore

have to overemphasize the role of the brittle part

of the lithosphere or fail to model discrete plate

boundaries accommodated by ductile, mylonitic

shear zones.

Additional weakening in the ductile level can be

obtained by treating also the energy equation, which is

neglected in the constitutive theory. We will therefore

refer to this relatively new theory as the ‘‘energy

theory’’ of localization. In a first step, following the

arguments of Hobbs and others (Backofen, 1972;

Hobbs et al., 1986), the theory for constitutive insta-

bility can, however, be recast to explore basic con-

ditions for energetic instability without solving the

energy equation. We will use this extended constitu-

tive theory as a lead in for reviewing the energy theory

of localization.

For the ductile flow, laws defined above potential

planes of localization are defined by the second

invariant of the deviatoric stress tensor and we can

write a suitable criterion for localization based on

effective scalar quantities (Appendix A)

strain hardeningdrVde

Vhcrit ð32Þ

A very similar formulation can be cast for the

nucleation of compaction or dilatation bands, when

replacing the effective shear stress by the pressure and

the effective shear strain increment by the volumetric

strain. For the high-pressure conditions, in geody-

namic problems, we do not need to consider such

pressure-dependent localization phenomena. Howev-

er, it turns out that hcrit is very sensitive to the

deviations from the smooth co-axial von Mises style

yield surface assumed so far. Geological materials in

the top 10 km can be described by a yield surface that

has corners (Coulomb envelope) and has non-coaxial

flow (strain-increment and stress vectors are not

collinear as in Fig. 3, see also Appendix B). Both

factors promote shear zones but especially the latter

ensures that the material localizes readily (Poliakov et

al., 1994). For the incompressible, non-coaxial case

the constitutive theory (Rudnicki and Rice, 1975)

predicts that bifurcations can occur for any amount

of strain hardening. At greater depth in the litho-

sphere, crystalline plasticity applies and flow becomes

co-axial. In the absence of yield vertices and non-

coaxial flow (Appendix B), the criterion for onset of

instability is negative strain hardening hcrit < 0 in the

time increment dt, i.e. the rock must become weaker

with deformation for shear zones to nucleate sponta-

neously.

We have already discussed the void-volatile

feedback mechanism as a potential mechanism for

strain weakening in the co-axial domain. It has been

argued that void coalescence can also lead to

departures from co-axiality, hence enhancing the

tendency for localization (Rice, 1977) according to

the constitutive theory (Appendix B). Propagating

void sheets may also significantly change the ener-

getics of deformation and localize as discussed in

the energy theory. This leads to the appearance of

additional, destabilizing (negative) terms in the

hardening law. A preliminary assessment of these

additional terms is that propagating void sheets act

destabilizing so that a conservative assumption is to

equate the additional terms to zero (Hobbs et al.,

1986).

Eq. (32) can be extended for the rate-sensitive

material as a suitable criterion for shear zone forma-

tion.

strain rate hardeningdrVde

¼ BrVBe

þ BrVBe

dede

Vhcrit

ð33Þ

Investigating the fluid dynamic visco-plastic for-

mulations laid out above, we find that they have zero

strain hardening (first partial derivate drops out) so

they should be good candidates for spontaneous

shear zone nucleation. However, all of these flow

laws have a positive strain rate hardening or zero

strain hardening in steady state. In a time-dependent

circumstance, the viscosity drops when creep is

accelerated but the stress still increases. By analogy

to the rate-independent solid, we expect that shear

zones do not form spontaneously in a smoothly

varying deformation field. Shear zones can still form

for strongly heterogeneous boundary conditions or

for negative strain rate hardening through spontane-

ous transitions from one flow law to another weaker

form.

An example for such a transition is shear zone

formation owing to grain size-sensitive creep from

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 309

mechanical standpoint (Braun et al., 1999). Dynamic

recrystallization relies on subgrain formation during

the movement of line defects in the power-law regime.

When the grain size is reduced sufficiently, a switch in

flow law can occur to a weaker grain size diffusion

dominated Newtonian viscous flow. At this point,

strain rate hardening can become negative and shear

zones can nucleate spontaneously. This transition is,

however, also thermally activated (Kameyama et al.,

1997) and a full thermal–mechanical energy assess-

ment is therefore necessary.

Leaving now the mechanical approaches and turn-

ing over to the thermo-mechanical shear zone models,

we have now to consider the energetics in a more self-

consistent manner. The shear heating term in the

energy equation and the thermal expansivity both

feed back positively into the momentum-rheology

equations. It is noteworthy that for increasing strain

or strain rate hardening shear heating also increases,

i.e. the vigor of thermal feedback increases. There-

Fig. 7. Principal feedback mechanisms and their validity field. A typical oc

In the brittle field, only the momentum-rheology feedback applies. The un

ductile field, a much richer choice of feedback mechanisms is possible thro

the mechanism in the brittle field, accelerated creep is possible through incr

p, water H2O and grain size g, and their feedback into the energy equation

brittle and ductile field. The additive strain rate decomposition allows natu

comprehensively coupled analysis is possible. Predominantly brittle or duc

the basis of numerical methods such as finite element analyses. Such behav

shown.

fore, for most materials, the contribution of tempera-

ture in the hardening law is negative (Backofen,

1972) thus promoting the spontaneous nucleation of

shear zones.

thermal � rheological weakening

drVde

¼ BrVBe

þ BrVBe

dede

þ BrVBT

dT

deVhcrit ð34Þ

Thermal–mechanical feedback is dependent on

the thermal scaling length introduced in Table 2. If

the deformation is fast (high strain rates), we obtain

a short scaling length on the order of the shear

zone thickness and we can discuss this in terms of

near-adiabatic conditions. For slow processes, heat

conduction plays a role. Heat conduction is a

negative feedback process and it will consequently

retard the onset of thermo-mechanical flow bifurca-

tion, or it will inhibit this kind of localization

phenomena altogether. Therefore, a fully coupled

eanic strength profile (Kohlstedt et al., 1995) is shown for reference.

derlying processes are summarized in Table 1. However, in the deep

ugh introduction of a thermal scaling length (Table 2). In addition to

emental changes of effective shear stress rV, temperature T, pressure

. Finally, shear zones can be triggered through coupling between the

ral mixing of brittle and ductile material properties and therefore a

tile material response and their interaction can thus be predicted on

iour does not need to be assumed as is done in the strength envelope

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349310

solution including momentum, rheology and energy

equations is necessary to comprehensively assess

the potential of generating shear zones at greater

depth.

2.8.2. Energy theory

The physics underlying ductile faulting can be

understood on the basis of a feedback diagram in

Fig. 7. In the constitutive theory, we only consider

the simpler feedback between the rheological and

momentum equations alone, with pressure p being

the feedback variable. This is sufficient for model-

ling faults in the brittle field. The energy theory

explains faulting in the ductile level and considers at

least one additional feedback variable, which is

either temperature T or new surface energy by ductile

cracking.

The energy theory of shear zone formation and

equilibration is the core subject of this review. It is the

key to understand ductile, mylonitic shear zones. Its

geo-scientific implementation is opposed to the clas-

sical constitutive, purely mechanical theory in the

subsequent chapters, where the weaknesses of the

latter are resolved in the subheadings on energy

theory. Energy-based approaches to localization have

been developed independently in the engineering and

the geodynamics community and have been rapidly

evolving over the recent years. The energy hypothesis

for shear zone formation has been put forward as a

rigorous theory (Appendix C) in the solid mechanical

engineering literature around 10 years ago (Cherukuri

and Shawki, 1995a,b; Sherif and Shawki, 1992; Zbib,

1992), while it has emerged in approaches to locali-

zation phenomena in fluid flow much earlier (Gruntf-

est, 1963). The theory has originally been restricted to

describe thermal feedback only, but recent extensions

also include the effect of surface energy (Bercovici

and Ricard, 2003; Lyakhovsky, 1997; Regenauer-

Lieb, 1999).

Unfortunately, in putting forward a new energy

theory, which is still far from being complete, there

has been little exchange between engineering and

geophysical approaches. This review focuses on the

recent advances in geosciences and therefore gives a

somewhat biased view. It is beyond the scope of this

review to unify the advancements, however, wherever

possible proper credit to the engineering community is

given (such as in Appendices B and C).

3. 1D shear zones

One-dimensional analyses have the advantages of

extremely high resolution, like hundreds to millions of

grid points, and provide an estimate of time scale. A

problem is of course the choice of appropriate bound-

ary conditions. Assuming to a zeroth order a shear cell

as shown in the inset of Fig. 2. We will discuss the

fluid dynamic approaches first, then proceed to the

visco-elastic and finally the elasto-plastic case studies.

3.1. Viscous thermal–mechanical feedback

A number of approaches have been formulated in

the 1980s that solve for fluid dynamic shear zones by

thermal–mechanical feedback in Newtonian and pow-

er-law fluids chosen as a proxy for lithosphere-as-

thenosphere conditions. Two end member boundary

conditions for the one-dimensional shear zone model

have been assumed: one in which the shear zone is

driven by constant velocity, the other in which the

shear zone is driven by a constant shear force (Fleitout

and Froidevaux, 1980; Locket and Kusznir, 1982;

Schubert et al., 1976; Schubert and Yuen, 1977,

1978; Yuen et al., 1978; Yuen and Schubert, 1977).

The equation of momentum conservation (Eq. (6)),

the power-law (Eq. (8)) and the energy Eq. (14) are

the only ingredients of the analysis. A small thermal,

compositional or velocity perturbation is assumed to

analyze the stability of basic shear flow. Since no

strain hardening was assumed it was found that shear

zones form readily under both boundary conditions.

Constant stress boundary conditions have been repeat-

edly found to lead to self-feeding thermal runaway

instabilities if the shear stress is high enough (Melosh,

1976; Spohn, 1980). Obviously, constant stress

boundary conditions require special settings in an

Earth-like scenario. We prefer to discuss runaway

instabilities later using a choice of more realistic

lithosphere rheologies and boundary conditions. How-

ever, the results of constant velocity boundary con-

ditions are perhaps more generally applicable in plate-

driven shear zones, and may give an insight into first

order processes in thermal–mechanical shear zones.

In Appendix C, we discuss that the constant velocity

boundary condition together with the condition for

thermally insulating boundaries also is a necessary

and sufficient condition for the existence of a homo-

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 311

geneous solution. This ensures that the localization

problem is well-posed and thus suited to investigate

the effect of rheology. The problem is free from

localization phenomena caused by a priori chosen

geometrical boundary conditions.

Under constant velocity boundary conditions, the

linear viscous system self-organizes into a stable

ductile shear zone with the following properties. A

maximum temperature is achieved inside the ther-

mal–mechanical shear zone whose magnitude does

not change with time but varies with rock type. The

thermal anomaly broadens conductively leading to a

change in width of the shear zone with the square root

of time and to an overall weakening of the system.

The most striking phenomenon is that, independent of

initial rock strength, the viscosity in the center of the

shear zone equilibrates to the same minimum. The

viscosity minimum in the center of the shear zone is

attained when shear heating and conduction are in

equilibrium giving the following thermal–mechanical

shear zone viscosity (Yuen et al., 1978):

Shear viscosity after feedback gmin ¼ 8jRT 2

max

Qu20

ð35Þ

For a stronger rock with a higher activation energy

Q, shear heating is more vigorous so that the maxi-

mum temperature in the shear zone Tmax is also

higher. This in turn leads, according to Eq. (35), to

the thermal–mechanical viscous strength compensa-

tion after feedback. Hence, the viscosity of the shear

zone is found to be controlled by the initial shear

velocity u0 of the one-dimensional shear cell. For

plate tectonic conditions, this thermal–mechanical

feedback viscosity is around 5� 1019 Pa s. For a

nonlinear power-law rheology, the same phenomenon

has been described and the shear zone also converges

to a quasi-steady state for which Eq. (35) gives an

approximate solution (Fleitout and Froidevaux, 1980).

Nonlinear viscous shear zones are narrower and have

a more realistic width of < 20 km.

3.2. Viscous thermal–mechanical and grain size

feedback

The analysis has been extended to include an

additional feedback between the momentum and rhe-

ology. This analysis is based on the observation of

extremely fine-grained shear zones in experimentally

deformed polycrystalline dunite (Post, 1977). It was

already noted that a switch in flow mechanism can

lead to strain rate weakening. The feedback is already

embedded in Eqs. (16) and (17). A reduced grain size

implies a lower flow stress, hence if the first term in

Eq. (17) outweighs the second term implying faster

grain size reduction than grain growth a condition for

mechanical instability is given.

Incorporating this feedback in addition to thermal–

mechanical feedback would lead to enhanced focusing

of shear zones into a width of several hundred meters

(Kameyama et al., 1997). The shear zone is stable

under constant velocity boundary conditions. The

important factor defocusing the shear zone is the

second term in Eq. (17) characterizing grain-growth.

It is obvious from Eq. (17) that shear heating implies a

grain growth and thereby diminishes the role of the

grain size sensitive mechanism. A careful re-investi-

gation of the grain size sensitive mechanism (Braun et

al., 1999) shows that the mechanism is probably not a

universal shear zone mechanism. In the last section

we will discuss, however, a jerky flow scenario where

grain size sensitive creep interchanged with thermal

heating pulses from ductile earthquakes can keep a

shear zone localized.

3.3. Visco-elastic approaches

We now investigate whether, for more realistic

rheologies, shear zones are unconditionally stable

under constant velocity boundary conditions or

whether runaway instabilities are possible. Thermal

runaway occurs, if the temperature increase owing to

shear heating leads to rheological weakening, feeding

back by increasing increments of shear heating. If the

feedback runs faster than the conduction can cool the

shear zone, an explosive heating phenomenon is

possible (Gruntfest, 1963). Thermal runaway has first

been suggested as a mechanism for deep earthquakes

by Orowan (1960). However, no quantitative proof of

the mechanism has been given until Ogawa (1987)

investigated a one-dimensional model of a visco-

elastic shear zone. He extended the above approach

by adding elastic strain rates via the additive strain

rate decomposition (Eq. (31)). The addition of elastic

deformation implies that the material around the

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349312

localized shear zone can act as a storage device for

elastic deformation to be released in the ductile shear

zone upon instability. Thus until the elastic energy is

used up, a situation that is akin to the constant stress

boundary conditions, can arise.

Using appropriate parameters for subducting slabs,

visco-elastic thermal runaway is possible under con-

stant velocity boundary conditions if the following

conditions are fulfilled (Ogawa, 1987): the shear

stress must be larger than 300 MPa, the strain rate

in the shear zone exceeds 10� 14 s� 1, the shear zone

width is smaller than 100 m and the viscosity inside

the shear zone is three orders of magnitudes smaller

than outside the shear zone. These conditions are not

outside a reasonable geological parameter range for

ductile—so called ‘‘mylonitic’’ shear zones (Handy,

1994). However, the required small width of the shear

zone has not been resolved numerically. Recall that

the power-law rheology used by Ogawa would predict

typical quasi-steady state shear zones with a width of

the order of a kilometer (Fleitout and Froidevaux,

1980).

Ogawa’s analysis has been extended (Kameyama

et al., 1999) to include the Peierls stress-regime (Eq.

(19)) in order to investigate whether the addition of

this mechanism promotes or stabilizes ductile ther-

mal–mechanical failure. When comparing the tem-

perature sensitivity of the Peierls stress mechanism to

that of the power-law, we find that at constant strain

rate the stress decreases with the inverse of the square

root of increasing temperature in the Peierls stress

case (Eq. (19)). This implies a close to linear temper-

ature–stress weakening relation in the Peierls stress

mechanism. In the power-law, on the contrary, the

exponential Arrhenius term (Eq. (15)) implies an

order of magnitude higher weakening for the same

temperature increment. In an additive power-law

Peierls stress rheology we would therefore expect

the Peierls stress to stabilize the thermal–mechanical

shear zone while the power-law would be prone to

runaway instabilities. This is in fact what has been

found out in the Peierls stress analysis (Kameyama et

al., 1999). The stabilizing effect of the Peierls mech-

anism is not strong enough, however, to prevent

thermal runaway under constant stress boundary con-

ditions. We will show in the discussion on 2D shear

zone models that with the additive strain rate approach

ductile thermal–mechanical instabilities are attracted

to the transition from Peierls stress-dominated to

power-law-dominated creep.

3.4. Elasto-plastic approaches

The shallow counterparts of ductile shear zones are

brittle fault zones. Brittle faults typically have a fault

width much narrower than their ductile counterpart.

Momentum-rheology feedback of non-associated

plasticity ensures individual faults narrower than a

meter scale, yet brittle faults can extend over several

tens of kilometers length. Although brittle fault zones

are not triggered by thermal–mechanical feedback,

the shear heating term in the energy equation still

holds. Furthermore, taking the high speed of a seismic

event into account conduction can be neglected. We

may speak of ‘‘quasi-adiabatic’’ conditions. It is not

surprising at all that brittle seismic events can lead to

melting on the fault plane (McKenzie and Brune,

1972).

In the one-dimensional approaches discussed so

far, strain hardening has been neglected. The above

analyses assume, without saying this explicitly, con-

ditions close to steady state. These are the conditions

for which the creep laws have been devised. When

looking at Fig. 2, we can see that shear zones most

likely nucleate during transient creep in the first bump

of the stress–strain diagram and they are fully estab-

lished under steady state conditions, i.e. the long

straight line of the diagram for which the creep laws

apply. Only very few analyses have been done with

the consideration of transient creep. This is due to the

lack of data on strain hardening. The linear stability

analysis laid out in the chapter on feedback has been

used with the upper bound estimate that the stress

after strain hardening cannot exceed Young’s modulus

(Hobbs and Ord, 1988). Even under these highly

hypothetic conditions, quasi-adiabatic shear zone for-

mation due to thermal–mechanical feedback is not

inhibited. We must still verify whether the adiabatic

condition represents a good enough approximation

and we should follow how the instabilities develop

nonlinearly through time. The finite amplitude re-

gime, of course, cannot be determined from a linear

stability analysis.

Recently, the potential for adiabatic shear zone

formation has been analyzed for geological conditions

using the power-law hardening model presented in

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 313

Eq. (12) in conjunction with the exponential temper-

ature term of Eq. (15). Again, the equation of mo-

mentum conservation (Eq. (6)) and the energy Eq.

(14) are used to solve for the dynamic evolution of the

shear zone. As a new element a new scaling length L

is introduced that is very much larger than the shear

zone width (Roberts and Turcotte, 2000). This new

scale serves as a description of the integrated elastic

area around the shear zone which can store elastic

energy. This elastic container adds an external stress

to the shear zone, governed by l the elastic shear

modulus of the elastic container and w the shear

displacement at the boundary of the slip zone. Hence,

the new force equilibrium is given by:

Force equilibrium rV¼ r0V� lw

Lð36Þ

where r0V is the initial shear stress. It follows that

under a constant velocity boundary condition a seis-

mic instability nucleates for a critical shear heating

(strain rate–energy density) or shear velocity. An

astonishing result is that the temperature increase

during the seismic instability is not large but only of

the order of 20 K (Roberts and Turcotte, 2000). The

integrated effect over a geological time scale is similar

to the results discussed for Newtonian thermal–me-

chanical shear zones.

4. Geodynamic modelling (summary of previous

work)

Geodynamic models of shear zones traditionally

have been separated into two different categories. One

group of modelling approaches is guided by observa-

tional data of geodynamic processes and the other

approach by the physics of the processes underlying

geodynamics. The former approach thrives at finding

a numerical method that adequately describes a given

observation, a top to bottom approach, while the latter

investigates fundamental modes of geodynamics from

the bottom up. In the former approach, a downscaling

scheme is used while the latter uses an up-scaling

theorem laid out in Tables 1 and 2.

This paper summarizes the theoretical framework

of modeling shear zones, i.e. it focuses on methods

that generate shear zones using the up-scaling scheme.

We describe briefly the inverse method of manually

inserting shear zones from observations. We feel that

the basic theory still deserves some more attention

before inverting for microphysical parameters from

large-scale geodynamical data sets. The future of

geodynamic modelling lies without doubt in a solid-/

fluid-mechanical approximation augmented by a

chemical equation of state for lithosphere and mantle

rocks. This needs to be compiled into a unified solid

earth reference database (Montesi and Zuber, 2002).

Investigation of shear zones from both angles of

attack would then be meaningful across different

disciplines and not only understood by few experts

who are aware of the significance of the basic as-

sumption in the simplified downscaling approaches.

We therefore restrict ourselves to reviewing the basic

developments in the field without specific application

to case studies. We will discuss in the second part only

one basic application, which is the quest for self-

consistent plate tectonics.

4.1. Viscous modeling

4.1.1. Constitutive theory

Viscous modeling of lithospheric deformation has

been introduced to understand the paradox of conti-

nental plates that deviate, when they collide, from the

basic paradigm of plate rigidity. The first analysis of

ductile plate tectonics was by Bird (1978), who

pioneered finite element techniques to model the

Zagros collision zone, which displays both bulk

shortening and a localized shear zone. For a success-

ful model run, Bird found that manual fine-tuning of a

low viscosity shear zone was necessary to appropri-

ately model the behavior of the Zagros crush zone.

This fundamental work was followed by investiga-

tions of the India Asia collision (England and McKen-

zie, 1982; Vilotte et al., 1982), where the ‘‘soft’’ Asian

lithosphere was modeled by a nonlinear viscous pow-

er-law fluid and the Indian indenter by a kinematically

driven boundary condition. Since the size of the finite

element mesh was very large (several tens of kilometer

mesh size or larger) and a fluid rheology was used

without feedback no discrete shear zones developed

self-consistently. The predicted results yield a

smoothed version of the observed deformation. Shear

zones have to be implemented manually. Rather than

using predefined zones of weaknesses an apparently

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349314

stiff inclusion, the Tarim basin (England and House-

man, 1985; Vilotte et al., 1986), can also be used.

Similarly, shear zones were not found as a natural

outcome of semi-analytical calculation of sinusoidal

stretching instabilities of a non-linear fluid lithosphere

(Fletcher and Hallet, 1983; Ricard and Froidevaux,

1986). Yet, shear band formation growing out of such

stretching instabilities is a classical experiment in

material sciences. Ductile shear zones are known to

form in materials that localize less readily than rocks,

e.g. metals. However, for numerical modeling of such

shear zones, some form of softening has to be consid-

ered (Needleman and Tvergaard, 1992). A fluid rhe-

ology without a yield like phenomenon or feedback

does not have the potential for forming shear zones. At

best, a fluid may be used as an up-scaled version of

discontinuous deformation with accepting the uncer-

tainty that some basic physics is missing. While in the

early 1980s, this approach was logical because com-

putational power was limited; nowadays, there is no

reason to use fluid approaches for the lithosphere.

Exceptions are cases where coupling to convection

in the mantle poses a large additional computational

workload (discussed below), or where a smoothing of

boundary conditions is the desired effect (Marotta et

al., 2001), or in laboratory experiments where similar-

ity criteria restrict the availability of analogue exper-

imental materials (Faccenna et al., 2001).

4.2. 2D visco-plastic modeling

4.2.1. Constitutive theory

Including a plastic limit stress by a pseudo-plastic

or a Bingham style formulation alone does not turn a

fluid model into a simulation with a shear zone

developing. Although strain hardening is zero, it is

not negative and the closest equivalence of a shear

zone occurs when strongly heterogeneous boundary

conditions lead to strongly heterogeneous shear flow

with some areas that hardly deform and others that

deform vigorously. This was in fact what was found

out in early calculations with a co-axial visco-plastic

formulation (Bird, 1988). When modifying the yield

phenomenon into a non-coaxial one, dramatic changes

are observed (Beaumont et al., 1996a; Ellis et al.,

1999; Lenardic et al., 2000).

Shear zones form readily in such a medium, the

only drawback being that non-associated plasticity

(Appendix B) applies to brittle fault zones only.

Because of the weak strength and the small thickness

of the brittle layer (Fig. 7), it is not an appropriate

model for the entire lithosphere. Another drawback of

this approach is that shear zones are rather fickle

features. New shear zones form readily during defor-

mation and old shear zones disappear altogether. A

way out of this problem is to self-lubricate shear zones

by parametrically imposed strain rate weakening laws

(Bercovici, 1993) or strain softening laws (Govers and

Wortel, 1995; Huismans and Beaumont, 2002). Both

formulations preserve shear zones once they are

formed. The advantage of the strain weakening over

the strain rate weakening law is that in the former

approach shear zones will retain their memory long

after deformation ceases, while in the strain rate

weakening case shear zones are instantaneous features

and vanish after a change of boundary conditions

(Zhong et al., 1998). One could argue that the para-

metric weakening curve records the history of feed-

back without any explicit modeling of feedback.

However, this would require a re-parameterization of

fully coupled feedback calculation, which has not

been done, to date.

4.2.2. Energy theory

Fluid dynamic calculations of visco-plastic shear

zones with coupled feedback phenomena have been

performed with varying degree of dynamic self-con-

sistency. Faults have been added as a frictional

boundary constraint to a viscous mantle (Zhong et

al., 1998) and the degree of shear heating close to the

boundary has been recorded in a kinematic model

(van Hunen et al., 2000). It turns out that shear heating

can accelerate deformation by about 20% through

lowering the viscosity down to 1020 Pa s. No dynamic

heating event was recorded in these predefined fault

zone models. A more significant effect has been found

for calculations in which a spasmodic release of

significant gravitational potential energy into shear

heating was possible (Schott et al., 2000). The shear

zone was left unconstrained. However, none of these

models was able to predict clear shear zones out of the

fluid dynamic approach. The image conveyed by

these calculations is that of highly transitional fea-

tures, which makes these approaches a poor candidate

for modeling plate tectonics. When considering void-

volatile feedback, the picture changes (Bercovici,

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 315

1998). Stable narrow shear zones generate self con-

sistently from pure volatile feedback.

4.3. 2D rigid-plastic modeling

4.3.1. Constitutive theory

Prior to the use of numerical models in geody-

namics, a simple analytical technique the so-called

‘‘slip line field method’’ has made a brief appearance

in geodynamic modeling of shear zones (Tapponnier

and Molnar, 1976). It is practically not used at present

apart from sporadic reappearance in the literature (e.g.

Regenauer-Lieb and Petit, 1997). The slip line field

method is only available for co-axial deformation with

simple plane strain or plane stress boundary condi-

tions. The rheology is simplified to a simple rigid-

plastic body. A static plastic equilibrium must exist,

i.e. the method cannot be extended to dynamic equi-

librium. Finally, being (semi-) analytical the solution

to problems of complex geometric boundary condi-

tions is tedious. Yet, prior to the availability of

numerical methods, the theory has governed 20 years

of metal deformation and has led to the rapid advance

of the theory of plasticity (Johnson et al., 1982).

Slip lines are the characteristics of hyperbolic dif-

ferential equations (see Appendix C). They are the

perfect mathematical embodiments of ductile shear

zones since they are capable of predicting either con-

tinuous shear or discrete shear velocity discontinuities,

i.e. vanishingly thin shear zones. What makes the

method so invaluable is that it is the only analytical

method that allows the prediction of shear zones in two

dimensions. For the plane stress case, displacement can

be three-dimensional. Since the existence of velocity

discontinuities simplifies the plastic solution tech-

nique, the strength of themethod relies on the weakness

of the numerical methods. In the engineering commu-

nity the method is therefore routinely used for bench-

marking new numerical techniques (Li and Liu, 2000).

Although the method does not, by definition, allow

for feedback it can be used to predict potential lines of

ductile failure that may or may not develop in a

ductile body with a less ideal rheology. Examples

are slip lines appearing as lines of dilatant fracture

(Coffin and Rogers, 1967), as lines of martensitic

transformation (Rogers, 1979) or slip lines appearing

as heat lines through shear heating feedback (Johnson

et al., 1964). The heat line approach has been used to

predict a time averaged maximum amount of shear

heating throughout the last 10 million years of colli-

sion in the Himalayas (Hochstein and Regenauer-

Lieb, 1998). The collisional energy dissipated on heat

lines was found to be sufficient to maintain the

observed anomalous heat transfer in convective geo-

thermal systems in the Himalayas. A large portion of

this energy appears to have been stored during initial

mountain building processes as gravitational potential

energy now released by extension (Tanimoto and

Okamoto, 2000).

4.4. 2D elasto-plastic modeling

4.4.1. Constitutive theory

In the case with elasticity, the tendency for local-

ized shear zones, so prominent in the rigid-plastic

approach, is wiped out significantly. This is due to

replacing infinitely thin rigid-plastic shear zones by

zones of finite width of an elasto-plastic material.

Although this analysis lends itself perfectly for the

investigation of realistic ductile shear zones, elasto-

plastic modeling of lithosphere deformation has tra-

ditionally been carried out without any focus on shear

zone development. Exception are analogue laboratory

studies (Shemenda and Grocholsky, 1992) or studies

of fault zones in the brittle field. An excellent review

of the studies in the brittle field can be found in

Gerbault et al. (1998). The prime interest of elasto-

plastic studies including the ductile field has been

focused on assessing flexural rigidity (see Albert et

al., 2000). Therefore, strain hardening/weakening and

feedback have usually been neglected. We will discuss

strain hardening and ductile elasto-plastic shear zones

in the section on applications.

4.5. 2D elasto-visco-plastic modeling

4.5.1. Constitutive theory

The lack of elasto-plastic studies on ductile shear

zones in the semi-brittle regime of Fig. 7 has been

rendered into the following simplified working hy-

pothesis (Burov et al., 2001; Cloetingh et al., 1999;

Gerbault et al., 1998; Moresi and Solomatov, 1998).

(1) Ductile shear zone nucleation processes are as-

sumed to be of second order importance. (2) The

propensity of shear zone nucleation in the brittle field

renders oblivious all other feedback processes. (3)

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349316

Hence, all shear zones nucleate in the brittle field. (4)

Shear zones propagate from the brittle field into the

ductile field and peter out. The most significant

drawback of this method is that the mechanical role

of the brittle layer is overemphasized. In Fig. 7, we

can see that the brittle zone is weak and presumably

does not extend deeper than about 10 km depth.

Thermal–mechanical feedback processes occurring

in the strong semi-brittle regime of the lithosphere

are completely neglected in these studies.

4.5.2. Intrinsic length scales and the energy theory

Elasto-visco-plastic modelling with feedback

includes all ingredients necessary for the investigation

of the transient creep phenomenon leading to the

nucleation of shear zones. The drawback of this

approach is that a proper implementation of the multi-

physics requires a large wealth of material data and is

computationally demanding (Tables 1 and 2).

The computational cost relies on the high degree of

spatial and temporal resolution that is required for

resolving the multiscale nature of the feedback. The

spatial scale that needs to be resolved before shear

zones are visible can be derived from one-dimensional

calculations and from theoretical considerations. In

the following compilation, we will see that geological

observations are much better suited to identify effects

of the different length scales than engineering appli-

cations. Through geological observation on shear

zones, we are in the unique position to pay back some

physical insight into the knowledge base compiled in

more than 50 years of theory of plasticity. Up to now,

the basic progress has been made in metallurgy.

Unfortunately, for metals, the intrinsic material length

scales of plasticity and thermal feedback (Lemonds

and Needleman, 1986) collapses into the micron-

scale, while in geology thermal feedback and meso-

scale plasticity spreads out owing to the slow defor-

mation and the low diffusivity of rocks. On the issue

regarding the nucleation of shear zones, a separation

of the length scales for shear zone formation is a

fundamental issue.

The intrinsic material length scale of deformation

by dislocations can be shown to govern the width of

shear bands in metals (see Aifantis, 1987 for a review).

The fundamental physics of this length scale hinges on

a breakdown of the classical continuum mechanics

where dislocation can be referred to as ‘‘statistically

stored dislocations’’ while below 10 Am the discrete

nature of dislocations is felt and there appear so called

‘‘geometrically necessary dislocations’’ which are re-

lated to the gradients of plastic strain in a material.

Recently, nano-indentation and micro-torsion experi-

ments have given support to this theoretically postu-

lated limit (Bulatov et al., 1998). It was found that it is

200–300% harder to indent at nanoscale than at large-

scale (see Gao et al., 1999 for a review). The imme-

diate outcome of this is that, in plasticity, there appears

an intrinisic material length parameter l1 characterizing

the energy of defects. This defect energy governs the

strain gradient of plasticity at mesoscale.

Material length scale of plasticity l1 ¼ MðlcnÞ2bð37Þ

where M is a material parameter (around 18 for

metals), l the elastic shear modulus and cn is a

reference stress coming in from the power-law hard-

ening law of Eq. (12) and b the Burgers vector which is

of nanometer scale. The strain-gradient plasticity the-

ory recovers at large-scale the power-law hardening

relationship when a macroscopic population of statis-

tically distributed dislocations is achieved. While this

length scale relies on the shear gradients, it has been

suggested to expand the theory to include a second

length scale for stretch gradients, which would govern

a critical void size before void-void coalescence (Fleck

and Hutchinson, 2001). All of these length scales are

below tens of micrometer scale.

Future analyses of shear bands should include strain

gradient effects mapping microscale dislocation inter-

actions into mesoscale cells (Guo et al., 2001). An

upscaling of these results then would allow to imple-

ment shear bands into large-scale shear zone models.

However, a clear separation of thermal feedback

(Lemonds and Needleman, 1986) and strain gradient

effects (Aifantis, 1987) has not yet been developed.

In geological applications, such a separation is

possible. The length scale l1 for rocks and ceramics

is also of micrometer scale but the thermal length

scale is much larger than in metals. In Table 2, we

have defined the thermal length scale to be:

Thermal diffusion length scale l2 ¼ffiffiffiffije

rð38Þ

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 317

This length scale is a new aspect of the energy

theory. It is the fundamental quantity controlling the

final postlocalization equilibration width of shear

heating controlled shear zones (Regenauer-Lieb and

Yuen, 2003; Sherif and Shawki, 1992) when heat

conduction and shear heating are in thermal–mechan-

ical equilibrium. It is also the quantity that governs the

resolution criteria for numerical thermal–mechanical

modeling of shear zones (Regenauer-Lieb and Yuen,

2003). In order to be able to see thermal feedback in a

numerical simulation, we need to resolve below the

thermal length scale. Taking e.g. a thermal diffusivity

of rocks j to be of the order 1�10� 6 m2 s� 1 and a

strain rate in the shear zone of the order of 1�10� 12

s� 1, we would obtain a thermal length scale of the

order of 1 km. This resolution is achievable in any 2D

simulation, even on a plate tectonic scale. The one-

dimensional models, discussed earlier, predict for the

case of a power-law fluid a thermally triggered shear

zone width of initially 2 km width widening with the

square root of time (Fleitout and Froidevaux, 1980).

We conclude that a 2D numerical approach in litho-

sphere dynamics needs to have a spatial resolution of

at least a kilometer, preferably smaller. The spatial

scale that is introduced by diffusion creep is the solid-

state chemical diffusional length scale, which depends

on the relative size of anions, such as silicates.

Chemical diffusion length scale l3 ¼ffiffiffiffiffiffiffiffiffiffiDeff t

pð39Þ

where Deff is the effective diffusivity at a given

pressure/temperature and t is the time. When this

length scale becomes important, diffusion accommo-

dated creep can become prevalent over deformation

assisted by dislocations.

This length scale critically influences the potential

for grain size feedback, hence is also an intrinsic

quantity of the energy theory of localization. In order

to resolve all the physics introduced by this feedback,

a numerical simulation would have to reach the scale

of the minimum grain size in the system upon which

diffusion can operate. Referring to Table 2, an upscal-

ing formalism has been suggested that describes the

flow of a statistical population of grains by a contin-

uum with a linear viscous flow law. In the calculations

of Kameyama et al. (1997), a spatial resolution of 1 m

has been reached. The predicted shear zone due to

feedback is in this case on the order of hundreds of

meters. Assuming that the smaller scale physics does

not change the behavior of the system, we conclude

that a 2D numerical approach needs to have a spatial

resolution of at least a hundred meters.

The spatial scale introduced by the void-volatile

feedback is on the order of a fluid inclusion (say 50

Am). This length scale can be introduced into strain-

gradient plasticity through consideration of an addi-

tional length scale from stretching strain gradients.

However, such a resolution is beyond reach for geo-

dynamic calculations but may be linked by discrete

particle method, such as smoothed particle hydrody-

namics (Monaghan, 1992).

Again, an upscaling scheme has to be used. Here

we assume normal void volume populations through

the population density parameters A and B in Eqs. (26)

and (27) for the ductile and brittle void nucleation

cases. Furthermore, in treating void volume as a

smeared continuum within a particular finite element,

any smaller scale physics is suppressed. Using finite

elements with a size of 200� 200 m, the void-volatile

feedback predicts relatively wide void sheets driving a

fluid-filled geodynamic shear zone with a width of

about 10 km (Regenauer-Lieb and Yuen, 2000b).

Ignoring possible feedback mechanisms at smaller

scale, we recommend a minimum resolution of 10

km for a void-volatile feedback calculation.

In sum, we would want to have a maximum

element size of the order of 100 m in order to be able

to resolve all feedback mechanism within a single

numerical analysis. Now, a typical 2D geodynamic

calculation would comprise an area of 1000� 100

km. This would imply about 10 million nodes in the

calculation. It becomes apparent why ductile shear

zones are hard to capture in geodynamical calcula-

tions. Ductile shear zones are however not beyond the

reach of current computers.

4.5.3. Energy theory

The toughest candidate for 2D shear zones is

undoubtedly the grain size sensitive feedback. The

only 2D calculation with grain size sensitive feedback

done so far focused on the physics of grain size

sensitive creep in convection. Therefore, it had a local

resolution of only about 5 km (Hall and Parmentier,

2003). This scale exceeded the spatial resolution

requirement for shear zone nucleation by an order of

magnitude. Consequently, in contrast to the 1D cal-

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349318

culations, no zones of highly localized deformation

were observed for obvious reasons.

Although the physics appeared to be fully imple-

mented, the first 2D calculations with thermal feed-

back (Chery et al., 1991) missed entirely the

phenomenon of thermal–mechanical shear zones.

The size of the finite element discretization was

chosen larger than l2. In order to be able to resolve

high strain rates and avoid undesirable mesh effects, at

least four elements should be contained within l2. The

implied 200 m resolution was implemented in an

idealized 2D setup of a perfectly homogenous iso-

thermal elasto-visco-plastic olivine plate under con-

stant extensional velocity boundary conditions

(Regenauer-Lieb and Yuen, 1998). Analogous to the

one-dimensional calculation a local perturbation in the

form of a weak inclusion was used (Regenauer-Lieb

and Yuen, 2000b). The schematic layout and the

predicted sinistral and dextral shear zones are shown

in Fig. 8.

We can now compare this elasto-visco-plastic 2D

calculation with the 1D elasto-plastic calculation of

Fig. 8. Isothermal Olivine Plate under constant plane strain extension. The

dimensional models a small weak imperfection was introduced as a nucl

power law flow law and void volatile feedback were considered. The shear

energy for 600,000 years. The same time lag has been reported in one-di

1994a,b). The void-volatile damage zone is trailing the thermal-mechanic

shear zone turns into a seismic event shown in Fig. 9.

Roberts and Turcotte (2000). Our formulation is in fact

the visco-elasto-plastic equivalent of the 1D calcula-

tion. The elastic scaling length L, which represented an

elastic container around the shear zone in the 1D

calculation, is implemented explicitly in the 2D calcu-

lations. It stores elastic energy during the charge up

time of 600,000 years during which about 12 km

elastic stretching of a 1000 km long elastic layer

occurred. Finally, the plastic threshold stress is reached

near the imperfection and the stored elastic energy is

released in seismic failure of the lithosphere. It is

surprising that only a moderate temperature rise of a

few tens of degrees is necessary to cause ductile failure

of the olivine sheet. Another important aspect is that

thermal runaway leading to melting instabilities is not

expected. In Figs. 9 and 10, we have plotted the results

of the thermal–mechanical calculations of Regen-

auer-Lieb and Yuen (2000b) and Roberts and Turcotte

(2000), respectively, showing the full evolution of the

seismic event. It is clear that the seismic event termi-

nates before 25 K shear heating have been achieved.

The addition of elasticity therefore has led to a dy-

olivine plate has dimensions 1000� 100 km. Analogous to the one-

eation point for shear zones. Only thermal mechanical feedback of

zone propagates rapidly through the plate after storing elastic strain

mensional models of ductile seismic instabilities in metals (Shawki,

feedback mimicking its crack like shape. After 800,000 years, the

Fig. 9. Temperature rise Tr due to shear heating versus time t. The

evolution of the thermal-crack like phenomenon is monitored close

to the imperfection. The top diagram shows a super-exponential

increase of temperature near the imperfection which commences

after 400,000–600,000 years of elastic loading of the system

depending on whether heat conduction is considered (non-adiabatic

case) or not (adiabatic case). The function Tr = a exp(exp(t)) has a

break in slope when the crack has propagated through the plate at

about 840 ka, but the temperature continues to rise dramatically.

Finally, the system turns into a seismic instability after a rise in

temperature of 10–16 K from shear heating for adiabatic and non-

adiabatic cases, respectively. The bottom diagram shows the

exponential reduction in stepping time of the thermally coupled

calculation. At the instability, the thermal feedback reaches times

steps of the order of seconds leading to a halt of the calculations.

Fig. 10. Temperature rise versus time for the 1D elasto-plastic

calculation (Roberts and Turcotte, 2000). Like in the 2D

calculations a super-exponential increase of temperature with time

is recorded. After 6 s, the instability goes seismic. The 1D

calculation continues through the ductile earthquake and records a

step up of temperature by about 24 K inside the shear zone after the

seismic event has stopped for a time larger than 10 s.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 319

namic instability, which after some time reaches the

same order of magnitude of shear heating as the purely

viscous shear zones.

In summary, we can infer that during the lifetime of

shear zones several feedback mechanisms play a

different role at different times. The first feedback

mechanism is the momentum-rheology feedback. An

elastic wave has been monitored (Regenauer-Lieb and

Yuen, 2000b) to propagate ahead of the thermal–

mechanical plastic wave which is shown in Fig. 8.

Depending on the energy stored in the elastic enve-

lope, the second thermal–mechanical plastic wave

turns seismic or not. Structural damage follows in

both cases mimicking a thermal–mechanical Mode II

crack-like feature. Structural damage occurs either

through void-volatile interaction or grain size reduc-

tion, thus engraving the shear zone for larger time

scales. In considering the multiphysics of shear zone

formation and their demand for spatial resolution

(below 100 m) and temporal evolution (below 1 s) it

becomes clear that 3D calculations are not yet riped

for any sensible undertaking, unless adaptive wavelet

methods are employed (Vasilyev et al., 2001).

4.6. 3D modeling

Three-dimensional calculations do not belong to the

classes of modeling discussed so far because they

attempt at solving a particular geodynamic problem

without going systematically through the physics of

the processes underlying shear zone formation. The

extreme spatial and temporal resolution demand posed

by the feedback calculations is circumvented by man-

ually introduced shear zones or by postulating simple

parametric rheological models or by considering only

numerically tractable feedback mechanisms. Three-

dimensional approaches to the fundamental problem

of self- consistent plate tectonics from mantle convec-

tion calculation are good examples (e.g. Tackley, 1998;

Trompert and Hansen, 1998) and shall be discussed

below. Other approaches focus on thermal–mechani-

cal feedback within the convecting mantle (e.g. Bala-

K. Regenauer-Lieb, D.A. Yuen / Earth-Sc320

chandar et al., 1995; Dubuffet et al., 2000) or attempt

to solve the problem of nucleation of shear zones in an

intraplate volcanic field by void-volatile interaction

alone (Regenauer-Lieb, 1999). A pioneering three-

dimensional shear zone model of the San Andreas

fault zone has been dealt with by Williams and

Richardson (1991) using visco-elastic rheology while

3D modeling of the plate-mantle interaction problem

has been first tackled with application to the Australia-

Antarctica subduction system (Gurnis et al., 1998).

For practical geodynamic purposes, simplified

approaches need to be developed. The key questions

that need to be addressed are: Is it possible to neglect

elasticity to suppress the tendency of the fully coupled

system to turn into a seismic instability, can strain

hardening be neglected, can plasticity be neglected? In

the following attempt at solving the plate tectonic

coupling problem, we will go through the different

approaches and show the importance of the individual

rheological ingredients.

5. Geodynamic modeling applications

Understanding plate tectonic coupling has been a

core question addressed in the geodynamic commu-

nity in the past 10 years. At the heart of this problem

lies the observation that plate boundaries are the

largest shear zones on the Earth (Gordon and Stein,

1992). They can last for hundreds of million years and

if they ever get inactive for some time they can be

reactivated at a later stage. What then causes the

nucleation of new plate boundaries? How can old

plate boundaries be reactivated? How can a plate

boundary survive for an extended geological time

period? How is the plate like motion coupled to

convection in the mantle?

The following discussion does not aim at giving a

review of the generation of plate tectonics from

mantle convection, but uses the ongoing discussion

summarized in review papers (e.g. Bercovici, 2002) as

a way to illustrate the fundamental limits of the fluid

rheological approaches that have been proposed. We

then focus on the basic problem of subduction initi-

ation for which solid mechanical models are available.

We use this as a common platform to discuss the

central role of elasticity, shear heating and water for

generating lithosphere scale faulting.

5.1. Visco-plastic plate tectonics

When looking at the long time scale of plate

tectonic cycles, it appears at first sight legitimate to

neglect elastic strains and only consider the role of

viscosity and plasticity. Implementing plasticity into

standard viscous mantle convection calculations

hence has been the main stream of attack. An example

for a 3D fluid-dynamic calculation (Trompert and

Hansen, 1998) that reproduced plate-like behavior of

the top surface by considering a Bingham-type rheol-

ogy is shown in Fig. 11.

The basic deficiencies of the model are immedi-

ately clear. The plate boundary is still diffused, i.e. no

discrete shear zone develops, there is only very little

vertical axis rotation (toroidal flow component is too

low), the strength of the lithosphere is too low, the

downwelling is two sided and the system does not

keep a permanent lithospheric identity. From time to

time, convective instabilities drag the entire litho-

sphere-like material into the mantle. The calculations

seem therefore more apt at describing a scenario that

has been postulated as a resurfacing event on Venus

(Grosfils and Head, 1996).

To improve some of these deficiencies, a system-

atic analysis of the yield stress has been performed

(Tackley, 2000a,b). In these calculations, the litho-

spheric yield envelope pictured in Fig. 7 was parame-

terized in a pseudo-plastic flow law. The pseudo-plastic

formulation rather than the Bingham visco-plastic body

allows increasing of the yield stress without going

suddenly into the stagnant lid regime. Recall that the

yield stress in the Bingham body (Fig. 4) acts as a

toggle switch between zero deformation below and

sudden deformation above the yield stress. The pseudo-

plastic law on the contrary allows some very small

deformation before the yield stress is reached (Fig. 5).

This minute detail is very important and allows a

range of coupling between mantle convection and lid

that cannot be observed in the Bingham approach. It

was found that the brittle strength contributed little to

the overall behavior of the lithosphere. Plate-like

results were achieved by a constant strength in the

ductile part of the lithosphere. If partial melting and

associated low viscosity asthenosphere allows for

additional decoupling of this stiff layer, a plate

tectonic scenario can be obtained self-consistently

(Fig. 12).

ience Reviews 63 (2003) 295–349

Fig. 11. Planetary resurfacing simulation with a Bingham visco-plastic flow law (Trompert and Hansen, 1998). A rather broad near-vertical

downwelling drags the cold surface layer into the mantle and leads to recycling of ‘‘lithospheric’’ material. The simulation is very similar to a

Venusian style tectonics where a cold stagnant surface layer is believed to be sporadically swallowed into the mantle in resurfacing events. The

Venus tectonic cycle is completed by cooling of the fresh surface layer, stagnant lid formation and renewed flushing instabilities.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 321

While this approach recovers more of an Earth-like

dynamics than the approach shown in Fig. 11, some

important shortcomings still remain. The yield stress

is higher but still too low when compared to labora-

tory analyses (Fig. 7). Also, the deficiency that

subduction is near vertical and has double sided

downwelling could not be resolved. Finally, no pure

strike slip faults exist. What these calculations clearly

show, however, is the importance of the yield stress of

the lithosphere. In the following we will focus on the

question how to destroy the integrity of the litho-

sphere and form a new plate boundary. Since spread-

ing centers seem to be well resolved by the above

visco-plastic calculations, we will home in on the

problem of subduction initiation as a key player in

Earth dynamics. Ultimately, we would also want to

abandon parametric approaches and merge them with

more complete rheological results from the laboratory.

5.2. Elasto-plastic passive margin evolution

As a first step we use a parametric power-law

elasto-plastic hardening model (Eq. (12)) and system-

atically vary the yield stress and the strain-hardening

power-law exponent. Elasticity is considered by cou-

pling elastic and plastic deformation in the Ramberg–

Osgood approximation (Branlund et al., 2001). The

Ramberg–Osgood approximation is a non-linear elas-

tic fracture mechanical approach that does not sepa-

rate plastic from elastic strain. The results are

compared to the additive strain-approximation (Eq.

(31)). Following an earlier suggestion we investigate

whether sediments loaded onto a passive ocean con-

tinent boundary (OCB) can cause failure of the

lithosphere (Cloetingh et al., 1982).

We have already discussed strain hardening in the

chapter on length scales. Strain hardening is a funda-

mental property of continuum mechanics communi-

cating microscopic discontinuous deformation at

nano-scale into a macroscopic plastic flow law. As

plastic strain increases, so does the dislocation densi-

ty. This leads to dislocation interaction, which in turn

is influenced by dislocation mobility. Metals can be

shown to have a power-law exponent that lies be-

tween 3 and 7 (Hirsch, 1975). Rocks and ceramics

have silicate covalent binding, which are difficult to

break. They have a high tendency for micro-brittle

failure at low temperatures. At higher temperature it is

easier to break the binding and the deformation by

dislocations increases. The importance of the strain-

dependent dislocation state in the temperature range

500–800 jC, has been neglected in the geological

literature. However, pioneering work by Griggs et al.

(1960) shows that strain hardening is very small for

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349322

Fig. 13. Model setup to test the influence of strain hardening on ductile failure of passive margins (Branlund et al., 2000). The lithosphere is

loaded incrementally by an increasing sediment load with a peak of 15 km after 100 Ma loading. The elasto-plastic lithosphere is supported by a

quasi-elastic foundation where the spring stiffness is reproducing the buoyancy contrast produced by displacing mantle material through

sediments and water.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 323

olivine. An adequate fit of the experimental results has

been obtained with a relatively high n of 35 (Branlund

et al., 2001).

The different hardening laws have been applied to

the passive margin model of Fig. 13 and a snapshot

after 60 Ma loading is shown in Fig. 14. The strain-

hardening exponent controls the stiffness of the plastic

response with a higher stiffness for higher n. This

analysis gives some insights into the effects of strain

hardening and it also shows that more realistic stress

levels for the wholesale failure of the lithosphere can

be achieved through dynamic interaction of elasticity

and plasticity. Although the elasto-plastic model gets

closer to the laboratory strength curves, it does not

quite reach the laboratory strength estimates. The

obvious solution to this problem is to consider viscous

deformation in the bottom part of the lithosphere.

Fig. 12. Plate tectonic simulations with a constant yield stress pseudo-plas

temperature iso-surface plot (right column). The left column shows a visco

red (highest viscosity). The system goes from distributed divergence with

spreading centers and optimum Earth-like torroidal flow at higher yield s

episodic rigid lid regime and finally at 340 MPa mantle convection is cov

5.3. Elasto-visco-plastic passive margin evolution

The idea of visco-elastic stress amplification as a

means to clip the high strength branch of the yield

stress envelope has been promulgated by Kusznir

(1982). The physics underlying visco-elastic stress

amplification is simple. Because the lower part of

the lithosphere can flow more readily than the upper

part it will, under an applied stress, deform by viscous

deformation. This consequently increases the elastic

stress field immediately above the flowing portion

until the yield stress is reached. Upon repeating this

process in the higher levels, the high strength branch

can be continually eroded and failure of the entire

lithosphere appears possible. The idea has been tested

for the case of passive margin evolution (Branlund et

al., 2001; Regenauer-Lieb et al., 2001).

tic lithosphere (Tackley, 2000b). The yield stress is indicated on the

sity plot with the color bar ranging from purple (lowest viscosity) to

localized downwelling at low yield stress (34–70 MPa) to sharper

tress (103–150 MPa) until at 220 MPa the system switches to an

ered by a stagnant lid.

Fig. 14. Zoom in of Fig. 13 showing various modes of lithosphere

collapse for different power law hardening exponents. The perfectly

plastic case is very similar to the power law hardening model with

an exponent of 35. An ideal yield stress of 200 MPa has been

assumed. Lithosphere failure is possible if the yield stress is lower

than 400 MPa. With increasing power law exponent the shear zone

becomes more focussed but the asymmetry of the sediment loading

function disappears.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349324

To illustrate this concept, the base model in Fig. 13

has been modified to incorporate the following effects.

The temperature profile of a cooling half space model

was added, the lithosphere has a composite visco-

elasto-plastic dry olivine rheology with diffusion,

Peierls and power-law creep incorporated by the

additive strain rate decomposition. The brittle top part

of 10 km is not considered, i.e. the sedimentary load is

immediately imposed onto the ductile portion of the

lithosphere. It was found that a fully coupled calcula-

tion localizes readily on any heterogeneity in the field.

This enhances the prospects for numerical grid arti-

facts. Therefore the singular peak of the sedimentary

loading function was smoothed and adapted to the

Western Atlantic passive margin. All nodal loads were

replaced by surface loads and the asthenosphere was

added as a viscous foundation. In order to investigate

whether asymmetric collapse is possible any source of

asymmetry other than the oceanic temperature profile

and the asymmetric loading function were removed.

Hence the geometrical heterogeneity at the OCB was

removed. A zoom-in on the highly deforming part of

the model is shown in Fig. 15.

Although visco-elastic stress amplification seems

to work it causes unexpected decoupled fluid- and

solid-like deformation, each with its own intrinsic

time-scale. Hence, it does not rupture the integrity

of the lithosphere. It is apparent that the lower part of

the lithosphere is too weak to deform as a solid entity

together with the upper part. Therefore, we obtain a

result, where the yield stress appears to be realistic but

for subduction initiation we still need to synchronize

the fluid and solid deformation in the lithosphere.

5.4. Add water

At this juncture, we may reconsider the two visco-

plastic models introduced in Figs. 11 and 12. The

model by Trompert and Hansen considers a Bingham

visco-plastic body and the model by Tackley a pseu-

do-plastic flow law. Both models reach the stagnant

lid regime at extremely different values of yield stress.

In the Tackley model it was found that the ‘‘best’’

results were obtained, if the lower part of the litho-

sphere had a constant yield stress thereby shielding

the lower part of the lithosphere from instabilities that

are shown in Fig. 15. By combining the Bingham

plastic model together with the pseudoplastic model,

we can expect that the lower yield stress of the

Bingham model shields the lithosphere from recycling

at low stress while the upper yield stress reaches the

excessive strength expected for a linear Bingham

viscous flow law at high strain rates.

Such a combined composite rheology is in fact

what is obtained by adding power-law and Peierls

stress mechanisms. The lower Bingham style yield

stress is embedded in the Peierls stress law (see Fig. 4)

and the high stress ceiling is part of the power-law

flow (see Fig. 5). Why are we not feeling the shielding

effect of the Peierls stress Bingham-like part in our

model calculations in Fig. 15?

The key in the success lies in the synchronization

of solid and fluid deformation. Solid mechanical

Fig. 15. Dry visco-elasto-plastic passive margin evolution (Regenauer-Lieb et al., 2001) showing contours of integrated strain. At 69 Ma, the

deformation, although having a higher degree of asymmetry bears some resemblance to the elasto-plastic case with high n (Fig. 14). Only

thermal–mechanical feedback is allowed, grain size sensitive and void-volatile feedback are not considered. A diffuse shear zone develops in

the upper part of the lithosphere but it does not propagate through the plate. It curves back to the surface. Fluid dynamic deformation at depth

occurs like a mirror image to the top deformation. The fluid style becomes apparent at 72 Ma when up and down-welling Rayleigh–Taylor

instabilities develop due to density contrast at the lithosphere–asthenosphere boundary.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 325

deformation in shear zones occurs at much faster

strain rates than the fluid style deformation in the

convecting mantle. The communication between shear

zones at surface and the fluid like deformation at

depth must be well coordinated to prevent a temporal

lag between deformation at surface and at depth as

seen in Fig. 15. If we increase the critical strain rate

for the onset of Peierls creep and thus shield the fluid

like layer of the lithosphere then fluid and solid

deformation may perhaps be coupled. The principal

parameter controlling this strain rate is the water

content in the mantle (Eq. (21)). The same model

calculation showed in Fig. 15 has been repeated by

raising the water content in the lithosphere (Fig. 16).

We can see from Fig. 16 that this logic holds. Just

by adding water, the shear zone can propagate through

the lithosphere instead of curving back to the surface.

Therefore, a new type of tectonics appears where

Rayleigh-Taylor-like instabilities at depth are shaped

by asymmetric shear zone in the top, which are, in

turn, fed by the gravitational potential energy release

of the negatively buoyant system. Coming back to the

feedback diagram in Fig. 7, there is a clear evidence

that water regulates the feedback between fluid-like

deformation at depth and solid-like deformation at the

surface. Water content in the lithosphere and in the

adjacent mantle dictates whether or not a lithosphere

scale fault can develop. It thereby regulates the style

of tectonics in a terrestrial planetary system (Regen-

auer-Lieb and Kohl, 2003).

6. Viscosity and lifetime of shear zones

A worldwide compilation of plate boundaries and

shear zones permits an inverse approach, allowing for

Fig. 16. Same as Fig. 15 but water has been added (COH= 810 ppm H/Si) (Regenauer-Lieb et al., 2001). Void-volatile and grain size sensitive

feedback are not considered. Solid and fluid deformations are coupled, and the lithosphere fails on its entire thickness. Ductile fault zones

develop dynamically. The major sinistral shear zone rotates counterclockwise during its evolution. A subsidiary sinistral fault is developing to

the left of the first hinge like shear zone. The top of the plate is weakened by zigzagging shear zones.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349326

the prediction of the long-term geodynamic strength

of plate boundary shear zones from observed plate

velocities. However, since shear zones have to be

implemented manually into the numerical approach,

they have been put in either through idealized velocity

discontinuities using a contact friction law on the fault

surface or through finite shear zones of lower effective

viscosity. The former approach has proven to be more

successful (Bird, 1998). A very low value of friction

of 0.03 was found on plate boundary shear zones

supporting the idea that plate boundaries are indeed

weak. More detailed regional analyses of the Africa-

Europe plate boundary along the Gibraltar–Azores

segment have given somewhat larger friction values

of the plate boundary (0.1–0.15) but it still appears to

be four times weaker than the adjacent lithosphere

(Jimenez-Munt et al., 2001).

While the mathematical idealization of a shear

zone by a velocity discontinuity with a sliding friction

law is a crude approximation the principal result of

weak shear zones cannot be disputed. We investigate

here whether the ductile shear zone in Fig. 16

becomes sufficiently weak. For this, we plot a viscos-

ity profile across the middle section of the major left

lateral shear zone in Fig. 16 (Fig. 17).

The shear zone is weaker than the model astheno-

sphere and is indeed weak enough to cause initiation

of asymmetric subduction. We conclude that consid-

eration of complete elasto-visco-plastic rheology with

thermal feedback resolves all of the deficiencies

reported in the above self-consistent approaches to

plate tectonics. We emphasize, however, that the

addition of water is just as important as thermal

feedback (Regenauer-Lieb and Kohl, 2003). Unfortu-

nately, for reasons of excessive numerical cost, these

high resolution calculations can at present only be

done in 2D. Surprisingly, the viscosity inside the shear

zone is of the same order of magnitude as predicted by

the simple one-dimensional viscous feedback calcu-

lations (Eq. (35)). This raises hopes of parameterizing

a simpler 3D approach with high-enough resolution,

which can be benchmarked by a complete 2D calcu-

lation.

Next, we discuss the lifetime of shear zones. We

have recognized the importance of shear heating in the

nucleation phase of shear zones. We have also seen

Fig. 17. Viscosity profile across the major left lateral shear zone in

Fig. 16. The shear zone has a viscosity minimum of 2.5� 1019 Pa s

and is several orders of magnitude weaker than the adjacent

lithospheric material (Regenauer-Lieb et al., 2001).

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 327

that shortly after the nucleation phase structural dam-

age will swamp the thermal–mechanical feature (Fig.

8). It is obvious that structural modifications are

guaranteeing the longevity of fault zones and their

potential for reactivation. When weighing in the two

structural mechanisms, i.e. void-volatile versus grain

size sensitivity, it is obvious that the void-volatile

mechanism is better suited for creating shear zones on

geological time scales. The diffusion of volatiles out

of the shear zones is very much smaller than the

healing through grain growth in grain size sensitive

creep or spreading of the anomaly through thermal

diffusion. This applies to carbon dioxide but not to

water because of the abundance of hydrogen related

point defects (Kohlstedt and Mackwell, 1998). The

bulky molecules of carbon dioxide in contrast will

remain trapped within the shear zones (Nakazaki et

al., 1995). This explains the observation of abundant

CO2 inclusions in xenoliths (Roedder, 1981). Hence,

we suggest to use water content as a global variable

and within shear zones consider only the void–void

interaction formulated in the section on volatiles.

Sheets with preferentially aligned CO2 voids in the

mantle are not the only factors that could guarantee

the longevity of shear zones. Other structural mod-

ifications have been suggested. Structural heterogene-

ity of the continental lithosphere (Tommasi et al.,

1995) or the mechanical anisotropy of olivine within

the mantle part of the lithosphere (Tommasi and

Vauchez, 2001) have been shown to preserve shear

zone memory and cause nucleation of shear zones in

preferred orientations. The longevity of shear zones

through macro-scale geological and mesoscale struc-

tural heterogeneity is a natural result of structural

geological observations.

They form a step up in scale of the discontinuous

processes discussed so far. Unfortunately no rigorous

formulation has been developed. There have been first

attempts at describing heterogeneous steady state

creep through their energetics. Consider, for instance,

a two-phase strength system, a dynamic evolution

towards an interconnected layer of the weak crystals

(Handy, 1994) during shearing, can embed a weak

fault into a structurally more competent host rock. The

dynamic evolution process of shear zone nucleation-

growth and coalescence of weak phases is formally

equivalent to the mathematical concepts of two-phase

flow introduced in the void-volatile mechanism (Ber-

covici et al., 2001a). We conclude that heterogeneity

is a prime candidate to produce and preserve ductile

shear zones on geological time scales. Macro-scale

structural heterogeneity evolves dynamically and

draws on the non-thermal energy fraction (1� v) ofthe deformational work stored inside the thermal–

mechanical shear zone.

The concept of heterogeneity brings us now to the

brittle field. Being potentially more heterogeneous

than the ductile part of the lithosphere, it is necessary

to look into the equivalent local approaches to fracture

also known as ‘‘damage mechanics’’. Damage is

stored as an additional internal variable and considers

the dynamic evolution of structural heterogeneity. We

have already looked into a ductile class of damage

mechanics, which we recommended to calculate semi-

brittle and ductile faults. In this case, the damage state

variable is the void–volume ratio (Eq. (22)). Brittle

faults can also promise longevity and an equivalent

state variable has been introduced that describes a

population of brittle microcracks. A complete damage

mechanical model for rocks has been developed by

Lyakhovsky and co-workers (Lyakhovsky et al., 1993,

1997). The advantage of damage mechanics over the

classical fracture mechanical approach is that it is

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349328

amenable to prolonged histories of brittle crustal

evolution with a complicated dynamic interaction

between local damaged zones, which are modeled

by zones of degraded elasticity. The disadvantages

are additional difficulties to formulate an objective

energy flow rate into the cracks and the problem of

mesh sensitivity. This approach, also dubbed

‘‘smeared crack’’ approach, has reached a high level

of sophistication in applications based on concrete

mechanics (de Borst, 2002). Following this line of

attack into an Earth-like scenario is a promising field

for understanding brittle fault zone dynamics.

Hybrid models that combine this method together

with discrete elements, modeling explicit cracks from

classical fracture mechanics, have been formulated

giving realistic fracture patterns. The success lies in

using discrete element and finite element methods

together, because the discrete element method consid-

ers discontinuous deformation and the finite element

model stores the continuum. This hybrid global–local

model unfortunately comes up with excessive com-

putation demands only to be realized in grand chal-

lenge computations with topline computers since 1

mm resolution has to be achieved to avoid mesh size

effects (Munjiza and John, 2002). The same applies to

particle codes discussed below.

The longevity and memory of fault zones remain

an unsolved problem in plate tectonics. While struc-

tural heterogeneity is the key to long living shear

zones it is not clear whether heterogeneity is embed-

ded primarily at brittle level thus repeatedly triggering

ductile shear zones in corresponding locations at

depth or whether it is caused by structural modifica-

tion in the ductile field itself. We would argue here on

the basis of the low strength of the brittle zone (Fig. 7)

and the proclivity of the strong semi-brittle layer to

nucleate thermal–mechanical shear zones that brittle

fault zones play a minor role in engraving plate

boundaries overlong time scales. They play, however,

an important role in the earthquake cycle.

7. Towards earthquake modelling

Comprehensive numerical approaches to earth-

quakes require a simultaneous solving of multiphysics

feedback processes at mm scale and a consideration of

the dynamic changes at large geodynamic scale. The

important issue of coupling tectonic and seismic length

scales has already been pointed out 10 years ago

(Anders and Sleep, 1992). In spite of the rapid evolu-

tion of computational power, we have not reached the

required temporal and spatial resolution to do this.

However, a strong economical push towards solving,

amongst other geodynamic phenomena, the earth-

quake problem has led to the development of large-

scale Earth computational projects such as ACcESS

(http://www.quakes.uq.edu.au/) investigating the ap-

plication of classical and new numerical techniques.

We will briefly summarize the current state of this

rapidly developing field with the most elementary

approaches.

7.1. Brittle models

One-dimensional brittle earthquakes models have

been formulated analogous to the ductile earthquake

model discussed earlier. A constant velocity is applied

to a simple spring–slider block system where the

sliding friction of the block replaces the ductile flow

in the shear zone. The friction law can be derived from

laboratory data giving a friction coefficient that is

dependent on the velocity and on the contact state,

e.g. gouge layer, between the sliding surfaces (Dieter-

ich, 1979b, 1992). Later work included also the effect

of shear heating (Blanpied et al., 1998; Chester and

Higgs, 1992) in the constitutional rate and state vari-

able friction model (Kameyama and Kaneda, 2002).

Kato (2001) showed that the shear heated fault goes

unstable at about 20% smaller pre-seismic sliding

compared to the fault without shear heating. The style

of instability is the same as shown in Fig. 10, i.e. from

observational data consistent with seismological stud-

ies of earthquake rupture characteristics, there is no

clear difference between a ductile and a brittle 1D

earthquake.

Two-dimensional models of brittle earthquakes

have also used a simplified Coulomb failure analysis

in which the dynamics of the rate and state variable

friction is neglected and the effective friction is gov-

erned by the Coulomb failure envelope. It is assumed

that the predefined fault remains locked during loading

until it reaches its failure criterion and then it fails

instantaneously. In between the faults, the material is

assumed to interact elastically. Both Coulomb and rate

and state variable friction models (Tullis, 1999) can be

Fig. 18. Discrete particle-dynamics calculation of a granular shear

zone (Mora and Place, 1998). A normal stress of 150 MPa is

mantained on the upper and lower edges of elastic blocks above and

below the region. The block is sheared as indicated by the arrows

and the deviatoric stress is monitored showing filamentary paths

with high stress. The model can explain the so-called ‘‘heat flow

paradox’’ in the brittle crust of the San-Andreas Fault and is a good

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 329

applied to data from modern earthquake catalogues,

however, there is no clear evidence of superiority of

one concept over the other (Gomberg et al., 2000).

While the potential of identifying faults from geodetic

observations and earthquake catalogues makes this

style of analysis appealing, the restriction to failure

on predefined faults without their capacity of devel-

oping smaller scale faults or dynamic evolution of

friction is a severe limitation. It has been shown that a

complex fault network displays dynamical modes not

observed in simple fault systems (Rundle et al., 2001).

The behavior of the entire earthquake fault network

system appears to self-organize in space and time into

particular modes that are also controlled by the inter-

action of changes in physics on the scale of single

faults and smaller (Ben-Zion and Sammis, 2003).

The fault network thus must be modeled as a whole

and the potential of fault zone propagation, degrada-

tion and healing must be built into the constitutive law

with considering a full coupling to the energetics.

Methods that allow just this have been presented for

both the brittle and the ductile field. They have been

found to be successful for describing the longevity

and memory of fault zones. An excellent discussion of

the brittle damage mechanics approach to model

single and network fault system has been given in

Lyakhovsky et al. (2001).

Another relatively new numerical approach has the

same potential. It is an up-scaled version of molecular

dynamics calculations. Rather than formulating the

mathematical problem in terms of a continuum it is

reduced to calculating the interaction between discrete

particles which when put together mimic the physics

observed at larger scale. Currently, friction, fracture,

granular dynamics and thermal–mechanical and ther-

mal-porous feedback have been implemented (Abe et

al., 2000). An example of granular dynamics modeled

with the particle approach is shown in Fig. 18. Since

the approach places itself at the lower scale of the up-

scaling scheme in the Table 1, it has the advantage

that micro-scale physics that are potentially over-

looked in the larger scale approaches are not

neglected. The obvious disadvantage is that geody-

namic scales cannot yet be reached owing to numer-

ical constraint. The approach is not restricted to the

brittle field but ductile shear zones can also be

modelled, using a discrete particle approach (Li and

Liu, 2000; Mora and Place, 1998).

7.2. Brittle versus ductile earthquakes

We have not discussed shear zone formation due to

phase transformations because of their restriction to

limited p–T conditions. Phase transformations may

not be capable of supplying a universal ductile earth-

quake mechanism but they may play a role in prepar-

ing conditions for deep ductile earthquakes (Karato et

al., 2001). However, we have pointed out that outside

the p–T conditions necessary for olivine-spinel trans-

formations there is already one important mechanism

for ductile earthquakes relying on thermal–mechani-

cal feedback. We have shown visco-elastic (Ogawa,

1987), elasto-plastic (Hobbs and Ord, 1988; Roberts

and Turcotte, 2000) and elasto-visco-plastic (Regena-

uer-Lieb and Yuen, 2000a) ductile thermal–mechan-

ical earthquake mechanism. The question arises as to

whether we can discriminate between brittle and

ductile earthquakes from observational data (Wiens

and Snider, 2001).

One important observational constraint is the direct

or indirect observation of heat released during an

earthquake or the cumulative heat released during

example for self-organized brittle network sketched in Fig. 19.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349330

prolonged seismic activity. The observation of negli-

gible heat flow anomaly over the San Andreas fault

zone at the Cajun Pass has been publicized as the

‘‘heat flow paradox’’ (Scholz, 2000a,b). The brittle

granular calculation of Mora and Place (Mora and

Place, 1998) has indeed proven that there is nothing

paradoxical about low heat flow in a granular

shear zone. We have pointed out that there is some

thermal mechanical feedback to be expected also in

the brittle field—in fact, it is possible to come up

with a frictional theory that relies on temperature

(Kameyama and Kaneda, 2002)—a good indicator

for brittle shear zones is their lower strength and

consequently also their lower heat release than their

ductile counterpart. We have pointed out that brittle

fault zones are also subject to degradation or healing,

hence, the opposite case of relatively high heat release

is no unequivocal evidence of ductile earthquakes.

Indirect evidence for ductile earthquakes can per-

haps be obtained from estimates of seismic efficiency

for cases of earthquakes with an exceedingly large

stress drop of the order of 100 MPa. Kanamori et al.

(1998) has shown that a lower bound assessment of

the energy released by the great Bolivia earthquake is

equal or larger than the energy released by the

eruption of Mt. St. Helens. Based on the present

knowledge of creep properties, it seems unlikely that

the Bolivia earthquake did not materialize into a

thermal–mechanical instability. However, did it start

owing to shear heating feedback and does the mech-

anism also operate for less extreme stress drop?

Another indirect evidence for thermal–mechanical

feedback is the observation of collocated deep earth-

quakes repeating in the same area within days (Wiens

and Snider, 2001). Thermal diffusion on a thin, meter-

scale, thermal–mechanical shear zone provides a

viable mechanism for repeating earthquakes. When

including thermal-elasticity into our numerical calcu-

lation for subduction initiation (Regenauer-Lieb et al.,

2001) we obtained thermal–mechanical instabilities

that comprise one element size showing that thermal–

mechanical ductile earthquakes are indeed expected to

have very narrow fault planes. We would like to point

out that only a modest amount of about 20 K (Roberts

and Turcotte, 2000) shear heating is necessary to turn

aseismic creep into a seismic instability. Therefore, we

conclude that ductile earthquakes belong as a natural

element to some mylonitic shear zones. Whether the

ductile instability turns seismic or whether there is just

a phase of accelerated creep (Ben-Zion and Lyakhov-

ski, 2002) depends on the temperature and material

parameter in the shear zone.

8. Summary

We have been discussing the basic numerical shear

zone concepts, their potentialities and their limits.

Thermal–mechanical shear zone formation has been

shown to rely on momentum– and thermal–mechan-

ical feedback processes. While the importance of

thermal–mechanical feedback in the brittle field is

weak, leading to the acceleration of the onset of

seismic instabilities, seismic instabilities or formation

of shear zones in the ductile field rely intrinsically on

thermal mechanical feedback fed by the exponential

dependence of creep strength on temperature. When a

shear zone has been fully developed, the relative role

of feedback processes changes. Deformational work,

dissipated prior to the formation of the shear zone in a

continuum around the shear zone, is now released

within the shear zone. This leads to important mod-

ifications of the energetics of the faulted system. We

have pointed out the implications of diverse time and

length scales.

On the large plate tectonic time-space scale, the

following characteristics have been derived. Mylonitic

shear zones take over the mechanical control of the

whole lithosphere. During the evoliution of the defor-

mation viscous modeling shows that mylonitic shear

zones become continually weaker, owing to the in-

creasing temperature inside the shear zone. This

temperature increase would go to a quasi-steady state

value that depends on the thermal properties of the

sheared lithology and its activation energy (Eq. (35))

and does not exceed 100–300 K reached after 10 Ma

shearing (Fleitout and Froidevaux, 1980). The width

of the predicted thermal shear zone (about 20 km) is

much larger than observed large-scale mylonite shear

zones of the scale of a few kilometers (Hobbs et al.,

1986). To resolve this discrepancy, other modifica-

tions of the energetics have been considered.

Grain size sensitive creep can only be efficient

under a narrow parameter range of shear zone cooling

(Braun et al., 1999). This is not possible with positive

shear heating but is workable during an intermediate

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 331

phase of cooling after a ductile earthquake (Fig. 10) or

if uplift or fluids cool down the shear zone. We can

assess the significance of grain size sensitive creep for

shear zone formation on the basis of a simple function

of the cooling rate for localization by grain size

sensitive creep given by Braun et al. (1999) in SI

units:

Cooling rate for grain size sensitive shear zones

log10T

k

� �¼ log10e þ 1:7 ð40Þ

where k is the constant defined in Eq. (17) and has a

value that lies between 10 and 20. Note that this

approximation has been derived by neglecting ther-

Fig. 19. Synopsis of shear zone observations in the field and inferred mat

sides of a network of brittle fault zone (top) and ductile mylonitic sh

intracrystalline plasticity at about 270 jC (van Daalen et al., 1999) marks

mal–mechanical coupling (Kameyama et al., 1997)

and the non-linear (power-law) aspect of the flow law

(Solomatov, 2001). However, when applying the

criterion to observed shear zones, the restrictive

nature of thermal–mechanical boundary condition

for localization by grain size sensitive creep becomes

apparent.

Individual fault segments inside mylonitic shear

zones have a width that lies well below 1 m (Drury et

al., 1991). Hence, if a width of 1 m were controlled by

thermal–mechanical conditions, we would imply a

strain rate for grain size sensitive creep larger than

10� 6 s� 1 (Fig. 19). From Eq. (40), we obtain a

cooling rate that must be of the order of 10� 3 K

s� 1. Such conditions are only possible after a ductile

erial properties and length scales for modelling to the left and right

ear zones (bottom). In the continental crust, the onset of quartz

the transition from fault to (semi-brittle) mylonitic shear zones.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349332

earthquake (Fig. 10). Ductile earthquakes, or phases

of accelerated aseismic creep >10� 12 s� 1 would

indeed predict sub-kilometer scale shear zones as

observed in the field. The magnitude of the heating

pulse during a ductile event (shown in Fig. 10 lies

well outside the conditions necessary to leave tell-tale

melts behind, so-called pseudo-tachylites). Grain size

reduction alternating with ductile earthquakes are a

viable explanation for large-scale networks of mylo-

nitic shear zones (Jin et al., 1998; Montesi and Hirth,

2003). This would imply a jerky flow at the scale of

the shear zone and the time scale of several thousand

years. Jerky flow is not uncommon in nature. For

instance, also found in metals at much smaller time

and space scales (Lebyodkin et al., 2001).

It still remains an open question as to whether the

small-scale thermal–mechanical conditions inside the

individual shear strands can control the large-scale

behavior of the entire shear zone. In order to resolve

this question, we need more powerful numerical

techniques that are able to resolve locally in centime-

ter scale and at the same time consider large-scale

geodynamic boundary condition at 1000 km scale.

Adaptive wavelet-based techniques (Vasilyev et al.,

2001) have the potential to do this and they may in the

future displace finite element approaches. Since the

thermal wave disappears over geological time scale

and is larger than observed large-scale mylonitic shear

zone (e.g. the Redbank shear zone in Australia (Hobbs

et al., 1986)) we have argued that longevity and

memory of shear zones must rely on additional non-

thermal storage of energy dissipated inside the shear

zone.

For a shear zone to become geologically perma-

nent, we should consider energy storage in terms of

new surface area as it is given by the nucleation of

volatile filled voids (Bercovici et al., 2001a; Regena-

uer-Lieb, 1999). Observations on volatiles released

from mantle shear zones shows that the maximum

width of the degassing zone is about 10 km (Rege-

nauer-Lieb, 1999). This volatile rich zone thus con-

strains the largest size possible for a mylonitic shear

zone. Experiments with rock analogues (Bauer et al.,

2000) give a good insight into fluid pathways inside

mylonitic shear zones and their role on dilatant plastic

evolution.

We conclude that for the purpose of modeling self-

consistent and self-organized plate tectonics we do not

need to go into a resolution of the shear zone network

as shown in the synopsis in Fig. 19 and resolved in the

numerical model for the brittle field in Fig. 18. A

realistic incorporation of the scale of dilatant path-

ways, engraving asymmetric weak structures,

inherited from elasto-visco-plastic deformation of

the lithosphere, would be enough for this purpose.

However, there remains the distinct numerical chal-

lenge of coupling fluid-like viscous deformation in the

mantle with solid-like elasto-plastic deformation in

the lithosphere. The calculation shown in Figs. 15 and

16 have been conducted with a solid mechanical code

which is too demanding for a global plate tectonic

calculation shown in Figs. 11 and 12. Fluid–solid

coupled multiphysics feedback calculation will have

important ramifications on understanding planetary

physics and contribute to resolve many open ques-

tions: Why has the Earth developed plate tectonics

and the other terrestrial planets have not? How do

volatiles, their release into the atmosphere and their

escape into space control tectonic cycles? What is the

role of shear zones on the surface of icy planets? What

controls the location, cyclic-like nature and dynamical

modes of earthquakes?

An understanding the life-cycles of shear zones

can currently be attacked from many different angles,

ranging from geological field studies for brittle (Petit

et al., 1999; Wibberley et al., 2000) and ductile

conditions (Christiansen and Pollard, 1997; Christian-

sen and Pollard, 1998; Tikoff et al., 1998), geochem-

ical studies (Downes, 1990), laboratory experiments

with rock analogues (Bauer et al., 2000; Bons et al.,

1993) and real rocks (Dieterich, 1979a; Mandl et al.,

1977; Post, 1977) as well as engineering applications

(Ananthakrishna et al., 2001; Fressengeas and Moli-

nari, 1987). We have outlined the recent advances in

numerical modeling of shear zones and have empha-

sized the multiscale physics of feedback mechanisms

that are important. In the synopsis we have pointed

out how geological observations of shear zone length

scales can be helpful in interpreting the basics of

shear zone properties with the aid of the simple

parametric laws that are obtained from numerical

modeling. The next step forward would be to go

beyond the specializations inherently grained in the

various fields and compile a truly multidisciplinary

dataset useful for lithosphere dynamics (Handy et al.,

2001).

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 333

Acknowledgements

We thank Bruce Hobbs, David Bercovici, Yehuda

Ben-Zion, Peter Mora, Hans Muhlhaus, Slava Solo-

matov, Shun-Ichiro Karato and Charley Kameyama

for stimulating discussions. This project has been

supported by both the geophysics program of the

N.S.F. and the Swiss Nationalfond 21-61912.00. This

is publication 1231 of the ETH Zurich.

Appendix A. From triaxial experiment to triaxial

flow law

A.1. Associated flow law

Because of the large experimental uncertainties, the

appropriate transformation of laboratory to tensorial

creep laws has only been discussed parenthetically

(Nye, 1953; Ranalli, 1995). If we assume that the

material is isotropic throughout flow and incompress-

ible, a generalized flow law can be expressed. We first

describe the classical purely plastic flow law:

eplij ¼ krijV ðA1Þ

where the superscript pl refers to plastic strain rate and

k is a function of position and strain history. In

plasticitiy theory, it is not a material property but a

scalar multiplier with dimension (s� 1 Pa� 1), which is

zero when the stress state is below the yield stress (e.g.

in the inside of the cylinder shown in Fig. 3) and some

positive value corresponding to the strain-dependent

hardening law allowing the cylinder to grow or shrink

as a function of strain hardening or weakening, re-

spectively. This is known as the Levy–Mises flow law.

It states that the stress and strain rates are everywhere

co-axial meaning that the principal axes of the stress

tensor and the strain rate are coincident. The flow law

furthermore implies that the components of strain rate

are proportional to components of the deviatoric stress

only and there is no pressure sensitivity. The classical

theory of plasticity does not consider time as a degree

of freedom and therefore the Levy-Mises flow law is

originally formulated with respect to the strain incre-

ment tensor instead of the strain rate tensor as illus-

trated in Fig. 3 in order to emphasize the time

invariance. In this figure, the principal of ‘‘normality’’

is also illustrated implying that the principal stress,

strain increment and strain rate axes, are normal to the

yield cylinder. Whenever we refer to this style of flow

law it is called ‘‘associated plasticity’’ synonymous

with ‘‘coaxial flow’’ or the flow is also said to be

‘‘normal’’ to the yield envelope.

In classical linear fluid mechanics, the same coax-

ial flow law is used but time plays a role, although

explicit time-dependent solutions can often be

avoided due to extremely slow, so called creeping

flow where the energy equation sometimes does not

need to be solved (see comments applied to creeping

flow in Appendix C). There is no yield criterion and kbecomes a true material property (the inverse of

viscosity), being constant for the simple Newtonian

fluid. Here, we are dealing with more complex flow

laws, which have a non-linear stress versus strain rate

relationship. In order to extend Eq. (A1) into an

associated flow law that has a non-linear stress–strain

relation, it is convenient to introduce scalar measures

of deviatoric stress and strain rate. Following Nye

(1953), we have defined an effective stress and an

effective strain rate, in Eqs. (3) and (5) accordingly.

Nye’s formulation is motivated by a pure shear plane

strain experiment in which the intermediate principal

strain rate is zero and continuity requires that the

maximum and minimum principal strain rates are

equal but have an opposite sign.

rV ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2rijVrijV

r

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

6

�ðr1 � r2Þ2 þ ðr2 � r3Þ2 þ ðr3 � r1Þ2

�s

ðA2Þ

e ¼ffiffiffiffiffiffiffiffiffiffiffiffi1

2eijeij

r

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

6

�ðe1 � e2Þ2 þ ðe2 � e3Þ2 þ ðe3 � e1Þ2

�s

ðA3Þ

the indices 1, 2, 3 refer to the principal stresses and

strain rates, respectively, and rijV is the deviatoric

stress tensor. Note that the effective stress and strain

rates are always positive. The above definition ensures

Fig. A1. A triaxial experimental setup is used for the determination of power-law and diffusion creep equation at steady state. The setup also

defines a standard for comparing other experiments (e.g. Vickers hardness test) by reporting the flow laws in terms of deviatoric stress versus

effective, maximum compressive strain rate.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349334

that the effective strain rate is equal in magnitude to

the maximum or the least principal strain rate measure

in the pure shear experiment.

It is straightforward to reformulate Eq. (A1) now

for a more generalized associated flow law where the

scalar factor is a function of the effective stress.

eviscij ¼ f ðrVÞrV

rijV ðA4Þ

where the superscript visc now refers to viscous strain

rates. For example, the power-law formulated in Eq.

(5) implies a tensorial viscous co-axial flow law

ePij ¼ a�nrVðn�1ÞrijV ðA5Þ

This viscous flow law turns into a visco-plastic flow

law if we define a yield threshold like in the Bingham

formulation (Eq. (8)). If we allow in addition elastic

deformation before reaching the yield threshold (Eq.

(A1)), we obtain an elasto-visco-plastic flow law.

Nye’s (1953) definition of effective stress and

strain rate is adopted in Ranallis textbook (Ranalli,

1995). We suggest to use a slightly different notation,

popular in the engineering community (Chakrabarty,

2000), which is motivated by triaxial conditions

depicted in Fig. A1 instead of the pure shear con-

ditions in the classical definition.

A.2. Triaxial experiment

Laboratory experiments usually report the creep

law in terms of differential stress versus uniaxial strain

rate in the piston direction of a triaxial experiment.

The experiment is shown in Fig. A1.

The principal strain rates are co-axial with the

principal stresses and their relative magnitude can be

obtained from mass conservation, if we neglect any

bulging deformation of the sample and consider the

equation of continuity.

e1 þ e2 þ e3 ¼ 0 ðA6Þ

Because of rotational symmetry of the experiment

around the maximum compression axis, the interme-

diate and least principal radial strain rates e˙2 and e˙3 are

of equal magnitude and it follows that their magnitude

is half the axial strain rate.

e3 ¼ e2 ¼ � 1

2e1 ðA7Þ

For ease of implementing laboratory data, we use,

however, a slightly different formulation based on the

von Mises equivalent deviatoric stress of the experi-

ment. These are obtained from Nye’s original formu-

lation of Eqs. (A2) and (A3) by a scalar multiplication

with the square root of three.

rV¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2rijVrijV

r

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2

�ðr1 � r2Þ2 þ ðr2 � r3Þ2 þ ðr3 � r1Þ2

�s

ðA8Þ

e¼ffiffiffiffiffiffiffiffiffiffiffiffi3

2eijeij

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2

�ðe1� e2Þ2þðe2� e3Þ2þðe3� e1Þ2

�s

ðA9Þ

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 335

Using the convention that compressive strain rates

and stresses are positive, we obtain the axial stress in

Fig. A1 as the maximum principal (compressive)

stress r1 and the radial stress of the confining medium

constitutes r2 = r3. It follows that the effective stress

for the numerical implementation can be calculated

from the laboratory deviatoric stress rD = r1� r3 as:

rV¼ ArDA ðA10Þ

By analogy, the effective strain rate can be com-

puted from the axial strain rate reported in the

experimental flow laws.

e ¼ 3

2Ae1A ðA11Þ

The only factor is thus the constant 2/3 for con-

verting laboratory flow laws into effective flow laws

for numerical calculations. This factor is valid for any

flow law. Note that Nye’s original formulation of

effective stress and strain rate based on pure shear is

awkward for rescaling triaxial experimental results

into effective flow laws.

This has also been noted by Ranalli. For power-

law, for instance, the strain rates have to be multiplied

by a factor of 2/(3(n + 1)/2) to transform triaxial devia-

toric stress–uniaxial strain rate equations into Nye’s

effective quantities (Ranalli, 1995). If this rescaling is

neglected, an increasingly large error is implied for

increasing n, e.g. one order of magnitude for n = 4.5.

For the Peierls stress, yet another scaling factor is

required, which is obsolete, if the above triaxial

definition of effective stress and strain rate is chosen.

Appendix B. Non-associated flow laws and

localization

For the mathematical treatment of mylonitic shear

zones, we have been dealing with associated flow laws

described in Appendix A for which the energy theory

of localization is required. We have, however, also

discussed a dilatant plastic material, which turns into a

strongly non-associated material when the brittle void

nucleation criterion is used (Eq. (27)). For details, see

Needleman and Tvergaard (1992); this paper also

provides an excellent review of the constitutive theory

of localization, which is a suitable criterion for de-

scribing localization phenomena in the brittle field.

In the following, we are giving a brief introduction

into non-associated plasticity, using the example of the

Mohr–Coulomb criterion. A very detailed review of

non-associated plasticity is found elsewhere (Vermeer,

1984). Subsequently, we briefly discuss the classical

bifurcation analysis and the development of a harden-

ing law that can lead to bifurcations. Note that the

constitutive theory of localization has not been devel-

oped to comprise localization of strain rate and ther-

mally sensitive solids (Rice, 1977). The following

discussion therefore interprets the flow law in terms

of a time-independent plasticity criterion only.

B.1. Non-associated plasticity and corners in the yield

envelope

If we neglect time-dependent quantities, Eq. (31)

simplifies into and elasto-plastic body

eij ¼ eEij þ eplij ðA12Þ

where we only add elastic and plastic strain rates. The

transition from a purely elastic to an elasto-plastic

state is given by the yield envelope. Eq. (23) gives an

example of an elasto-plastic pressure and deviatoric

stress-dependent yield function U which collapses

into the von Mises envelope for q1,2,3 = 0. The von

Mises yield envelope is illustrated in Fig. 3, which is

the basis for definition of the scalar multiplier k in the

associated Levy–Mises flow law, i.e. k = 0 inside the

von Mises cylinder and on the cylinder k>0.

U ¼ rVryV

� �2

�1 ¼ 0 ðA13Þ

For a von Mises solid, the direction of flow is

normal to the yield surface, so the flow potential

coincides with the yield envelope and the flow law

can be expressed as a function of the effective stress

only. Extending the flow law into a generalized plastic

flow law where flow and yield potential may not

coincide, we rewrite Eq. (A1) into

eplij ¼ kBG

Brij

ðA14Þ

where G is the flow potential, giving the direction of

flow after yielding. If G =U, the flow is associated but

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349336

for inequality, we speak of a non-associated flow, i.e.

the flow is not normal to the yield envelope. An

example for non-associated flow is the Mohr-Cou-

lomb plastic body, which is here expressed in terms of

principal stresses (Vermeer, 1984):

U ¼ 1

2ðr1 � r3Þ þ

1

2ðr1 þ r3Þsinl � c cosl ðA15Þ

where l is the friction angle from the Mohr–Coulomb

failure envelope and c the cohesion. This yield enve-

lope has corners when plotted in the three-dimension-

al stress space. Another feature of the Mohr–

Coulomb failure envelope is that it does not depend

on the intermediate principal stress as shown in Eq.

(A15). Similar to the yield envelope the flow potential

does not depend on the intermediate principal stress

but depends on the dilatancy angle b instead of the

friction angle l.

G ¼ 1

2ðr1 � r3Þ þ

1

2ðr1 þ r3Þsinb þ const ðA16Þ

if the dilatancy angle b is equal to the friction angle l,the Mohr–Coulomb law turns into an associated flow

law. This condition is, however, too restrictive (Rud-

nicki and Rice, 1975).

Interpreting the Mohr–Coulomb flow law in the

framework of the triaxial experiment (r2 = r3) shown

in Fig. A1, it becomes immediately apparent from

substituting r3 by r2 into Eq. (A16) that we have to

deal with two potential functions and two scalar

factors owing to a corner on the yield envelope.

eplij ¼ k1BG1

Brij

þ k2BG2

Brij

ðA17Þ

It is obvious that rheologies with such singular

transitions in flow laws are prone to localization on

preferred planes.

B.2. Localization bifurcations

For a quantitative investigation of these instabil-

ities, bifurcation analyses have been done (Needleman

and Tvergaard, 1992; Rice, 1977) and critical hard-

ening has been predicted as a basis for a constitutive

theory of localization. In such analyses, conditions for

band like instabilities within homogeneous, homoge-

neously deforming rate-independent solids are derived

and a correspondence for the occurrence of stationary

body waves has been found (Rayleigh and Stoneley

waves). Characteristic directions of localization are

found to be preceded and guided by these elastic wave

phenomena. Bifurcations are associated with a loss of

ellipticity in rate-independent solids. However, for

rate-dependent solids, the consititutive theory finds

that localization bifurcations are effectively sup-

pressed, i.e. the governing equations remain elliptic.

The energy theory of localization provides further

insight for the case of ductile shear zones.

Appendix C. Energy theory of localization

As an additional element, the energy theory of

localization takes the modification of the local energy

during deformation into account. This is a very

important step up in physics and requires marriage

of classical mechanics and non-equilibrium thermo-

dynamics. Note that both the continuity and the

momentum equations are independent of time. Time

dependence only arises through the energy equation.

We have already pointed out that, in geodynamics and

engineering applications, we can often use a quasi-

static (plasticity theory) or the so called creeping flow

regime (fluid mechanics), where it is possible to

sometimes ignore the effect of time. However, when-

ever throughout deformation, there appear time deriv-

atives in the local energy quantities, the time

invariance must be abandoned. This applies to both

solid- and fluid-mechanics, although it appears to be

more naturally accepted in the mantle convection

community, for which reason most of the early energy

concepts to localization, reviewed here, have been

dealing with fluid rheologies.

It is obvious that solids display some important

modifications in the local energetics after shear zone

formation. It follows that the energy theory is required

for defining the shear zone width after localization.

The lack of a physical scaling length governing shear

zone width is one of the shortcomings of the classical

constitutive theory of localization. We will show that a

critical local energy density acts as a trigger to

localization, thus overcoming the elliptical solution

space for ductile rate-dependent solids, which is the

second most spectactular failure of the classical con-

stitutive theory of localization.

rth-Science Reviews 63 (2003) 295–349 337

C.1. Thermodynamic criteria for instability

For this, we look at the internal energy of a homog-

enous volume element of a sample, which in classical

equilibrium thermodynamics is characterized by n + 1

state variables, where temperature is the variable for

n = 0. For the time being, we assume that there is no

flux of energy from external sources through a radia-

tion term. It has been argued that this is not a necessary

restriction since the radiation term does not determine

the thermodynamic process but, rather, the thermody-

namic process determines it (Lavenda, 1978). The book

by Lavenda gives a critical review of the theory of non-

equilibrium thermodynamics from the time of birth of

the chaos theory. We recommended this book as a

further reading on the basic concepts. In the following,

we only give a brief introduction to the theory.

During mechanical deformation thermodynamic

state variables, governing mechanical properties com-

prise first of all the elastic strain and the absolute

temperature. During deformation, additional micro-

structural variables come into play, which can char-

acterize dislocation density, phase changes, damage

(new surface energy), etc. These are often expressed

in terms of tensorial functions of strain rate energy

densities (product of stress and strain rates). We have

discussed that before flow bifurcation the system

deforms homogenously so that non-equilibrium local-

ization phenomena can be seen as a dynamic sequence

of evolving thermodynamic equilibrium states. Taking

one particular equilibrium state at time t, we write the

specific Helmholtz free energy w of this volume

element as a function of its n + 1 state variables

wðT ; eE; aiÞ; 1VjVn ðA18Þ

where the elastic strain eE and the absolute tempera-

ture T are the first two state variables a0,1. The secondlaw of thermodynamics leads to the inequality of

Clausius–Duhem

� DwDt

� q

TrT ¼ rijeij � q

BwBaj

DajDt

� q

TrTz0

ðA19Þ

where q is the heat flux vector out of the reference

volume and the term including the material time

derivative Daj/Dt gives the stored energy terms, which

appears for instance in new surface energy during

K. Regenauer-Lieb, D.A. Yuen / Ea

microcracking (Chrysochoos and Peyroux, 1997).

Now, the double product of the Cauchy stress tensor

and the strain rate tensor gives the mechanical power,

which also contains the non-dissipative reversible

elastic deformational work rate (so-called isentropic

power). We subtract this work out of the entropy

change in Eq. (A19) by the first material derivative

of the specific Helmholtz free energy for n= 1. We

already would like to point out here that an additional

feedback term (comprising the second derivative)

appears later in the derivation of the energy equation.

By analogy, all other stored energy terms for higher j

can be subtracted likewise. In order to assess the

dissipation out of equilibrium, we define the intrinsic

dissipation function Ri and perturb the equilibrium

system by small velocity perturbations ni.

1

2Rininiu� Dw

Dt¼ rijeij � q

BwBaj

DajDt

ðA20Þ

Eq. (A19) embodies the core of the energy theory

of localization, mathematically expressing the dissi-

pation as the sum of force-flux products where within

each product there appears a state variable. While in

classical plasticity, the system is considered mathe-

matically closed when the conservation laws of mass

and momentum are satisfied, in the energy approach

the additional consideration of the energy fluxes in the

specific entropy production associated with Ri gives a

closed system. This opens the way to non-elliptical

solutions essential for the phenomenon of localization

(Aifantis, 1987) as we show below.

A necessary but not sufficient condition for stabil-

ity is that the system dissipates positively. This

implies that in a generalized Gibbs space, spanned

by all independent velocities, the dissipation function

must be given by an ellipsoid centered on the origin.

Following states can be distinguished in a generalized

velocity space (Lavenda, 1978):

Ellipsoid; for all i Ri > 0

Parabolic; at least one Riz0

Hyperboloid; at least one Ri < 0

ðA21Þ

The ellipsoid space is a necessary but not a sufficient

condition for a homogeneous solution. Within the

ellipsoid solution space geometrically controlled shear

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349338

zones are possible, for instance. We will not deal with

such shear zones that are predefined by geometry. The

parabolic regime, on the contrary, is a sufficient con-

dition for material instability and it is said to be in a

Ellipsoidal

Paraboloid

Hyperboloid

Pu

Be

Ri>0

Ri=0

Ri<0

Fig. A2. Localization criteria and their expression in a pure shear experi

Without feedback ductile, deformation takes place in the elliptic regim

preferred slip planes arises when one of the Ri’s, related to a single therm

mathematical system to calculate these bifurcations. Intrinsic length scal

parabolic regime, i.e. shear bands have a finite thickness. There is no mesh

In the hyperbolic regime, shear zones become slip lines, i.e. they are ma

characteristics (Dewhurst and Collins, 1973) is a suitable mesh-less soluti

state of meta-stability. In thermodynamics, this is also

called a ‘‘marginally stable state’’. It follows that a

single internal state variable linked to a source of

internal power can cause flow localization (Fig. A2).

re Shear Experiment

fore After

Homogeneous Deformation

Slip Lines (velocity discontinuities)

ment. Ri is the intrinsic dissipation function defined in Eq. (A20).

e where homogeneous deformation persists. Shear localization on

odynamic state variable, is zero. The energy theory offers a closed

es, defining critical energy levels, ensure that solutions stay in the

sensitivity if the numerical resolution matches intrinsic length scales.

thematical idealization with a vanishing thickness. The method of

on tool for these idealized rheologies.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 339

While this terminology defines flow localization

within the framework of the energy theory, we also

would like to be able to predict post-bifurcational

evolution of the shear zone. The conservation law

of power introduced in the following paragraph

allows the incorporation of feedback terms, while

the variational principle of least dissipative power

gives post-bifurcational time-dependent evolution,

which is self-consistent in terms of classical me-

chanics and compatible with non-equilibrium varia-

tional thermodynamics. This enables us to formulate

criteria for numerically tractable solutions in the

post-bifurcational state using the variational princi-

ples of finite element analyses and an adaptive time

stepping scheme controlled by a critical thermody-

namic meta-stable state (Regenauer-Lieb and Yuen,

2003).

C.2. Energy equation

Consider the same volume element in thermody-

namic equilibrium. We have been describing its inter-

nal power in motion using the Lagrangian, also called

the substantial or material time derivative implying

that in our mathematical description we are moving

with the deforming volume element. In fluid mechan-

ics, this volume element is sometimes called a fluid

parcel. Integrating with respect to time, we thus obtain

the specific internal energy eint of the reference

volume/parcel in motion and considering its kinetic

energy ekin by inertia we obtain with the classical

mechanical energy balance.

ZV

qetotdV ¼ZV

qeintdV þ 1

2

ZV

qv � vdV ðA22Þ

where v is the velocity vector and V the reference

volume. We are interested in how this energy changes

with time so in the following we always consider the

substantial/material derivative and the law of energy

conservation turns into a conservation law of power.

Now, in geodynamic deformation, we mostly deal

with negligible kinetic energy, i.e. we use the quasi-

static/or creeping flow approximations and set the

kinetic energy to zero. This does not apply to earth-

quake mechanics. Additionally, we simplify the equa-

tions by omitting the volume integration, i.e. we

always assume an arbitrary reference volume in ther-

modynamic equilibrium.

Detot

Dt¼ Deint

DtðA23Þ

The thermodynamic energy balance for the specific

energy is given in terms of entropy s by

eint ¼ wðT ; eel; ajÞ þ sT ðA24Þ

additionally

s ¼ � BwBT

ðA25Þ

and

Ds

Dt¼ � B

2wBT2

DT

Dt� B

2wBTBaj

DajDt

ðA26Þ

where the specific heat ca is defined as

cau� TB2w

BT2ðA27Þ

In the development of the specific Helmholtz free

energy of the reference volume, we have assumed that

the flux of power r by radiation is zero. We now relax

this condition and write the basic balance of power,

which in continuum thermodynamics is given (Green

and Naghdi, 1965)

ZV

qDeint

DtdV ¼

ZA

qdAþZA

rdA ðA28Þ

where A now stands for the surface area of the

reference volume V and outwards directed flux is

positive. Substituting Eqs. (A24–27 into Eq. (A28),

we obtain

ZV

qDwDt

þ qT � 1

Tca

DT

Dtþ B

2wBTBaj

DajDt

� �dV

¼ZA

qdAþZA

rdA ðA29Þ

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349340

Using the intrinsic dissipation defined in Eq. (A20)

and rearranging terms the balance of energy now

reads:

ZV

qcaDT

DtdV ¼

ZV

rijeij � qBwBaj

DajDt

þ B2w

BTBaj

DajDt

dV �ZA

qdA

�ZA

rdA ðA30Þ

When comparing Eq. (A30) with Eq. (A19), we

note a new, third term on the right side of Eq. (A30).

This gives the additional coupling term in the energy

equation, while the last term including r is the external

source term through e.g. radiation, chemical reactions

and Joule heating, etc.

The energy Eq. (A30) is completely based on

thermodynamic state variables; we will now go on

and simplify. This is done by considering, what we

appreciate to be, the most important effects. Note that

there is no current consensus on the role of elasticity

between fluid and solid-mechanical communities and

the feedback owing to the creation of new surface

energy (void creation) may be considered more im-

portant than the effect of elasticity. We have argued in

this review that both terms are important in the ductile

regime. We would like to point out that the theoretical

thermodynamic framework for localization by ductile

damage through void creation summarized here is

severely simplified. A more complete thermodynam-

ically inspired theory of localization due to void

creation, based on the energetics of two phase flow,

can be found elsewhere (Bercovici and Ricard, 2003).

This theory is tailored to describe brittle processes,

without considering elasticity. However, it overcomes

the deficiency of mesh-dependent solutions inherent

in the classical elasto-plastic constitutive theory of

localization.

C.3. Simplified energy equation

We now simplify the energy equation by specify-

ing again an arbitrary volume in thermodynamic

equilibrium and spelling out the thermodynamic quan-

tities in terms of physical ‘‘constants’’, which are

certainly not constant but, themselves, dependent on

thermodynamic state variables as demonstrated in the

above definition of the specific heat.

First of all, we want to separate out the elastic from

the visco-plastic work because the former appears as

stored energy and the latter, as the source of heating.

The stored energy gives rise to an additional coupling

term in the energy equation, describing the interaction

between temperature and the other thermodynamic

state variables. The first state variable has been

introduced as the elastic strain and this coupling term

consequently describes the thermal-elastic effect.

Thermal-elasticity takes into account that the material

dilates on heating and shrinks on cooling. Another

important coupling could be latent heat release upon

phase transitions. In the following, we simplify the

energy equation by only writing down the elastic

coupling term, which is also known as isentropic

power by adiabatic volume changes being

qTB2we

BeEijBTeEij ¼ kthTequ

Dp

DtðA31Þ

where kth is the linear coefficient of thermal expansion

and Tequ is the equilibrium temperature change of

adiabatic expansion/compression.

We use the additive Maxwell body decomposition

(e.g. Eq. (A12)) and note that we can separate out

visco-plastic from elastic power by:

rijeij ¼ rijeEij þ rije

viscij ðA32Þ

We have already discussed the influence of the

elastic power, now we will discuss the second term,

the double product of visco-plastic strain rates and the

stress tensor, giving the dissipative power.

rijeviscij ¼ vrijVe

viscij þ cðp� pvÞ

1

q2

BqBt

ðA33Þ

where we separate out deviatoric from isotropic dis-

sipation processes. The prefactors give an additional

simplification by dropping the stored energy terms

(e.g. surface energy due to dilatancy which would go

into the second term on the right in Eq. (A30)) and

lumping them into a scalar factor 0 < v, c < 1 thereby

diminishing the shear heating term or the dissipation

through the volume change, respectively. The pressure

stress pv is due to bulk viscosity causing the total

dissipative volume change. This term is often consid-

ered in extended Boussinesq approximations of man-

tle convection (Yuen, 2000) but for the purpose of

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 341

faulting in the ductile part of the lithosphere it can be

conveniently neglected. The energy equation is now

simplified to

qcpDT

Dt¼ vrijVeij þ kthTequ

Dp

Dt� qcpjj

2T ðA34Þ

where the conduction is spelt out in terms of diffu-

sivity j (note that, for large strain conduction, it is

also strain-dependent (Povirk et al., 1994) and not

necessarily isotropic).

C.4. Simplified energy theory of localization in Earth

and engineering sciences

While the constitutive theory appears to have

reached a stage of maturity, the energy theory defi-

nitely has not. In particular, the vexing separation of

the different scientific communities and the differ-

ences in notation of geo- and engineering style has

prohibited its level of acceptance. Perhaps, the most

radical drawback in Earth Sciences is that the brittle

solid observed at surface localizes readily, so that little

effort has been devoted to investigating an appropriate

theory for the ductile level. This is not the case for the

deformation of metals and there has been considerable

effort in trying to understand localization phenomena

in metals, which cannot be explained by the standard

constitutive theory.

An excellent summary leading to the formulation of

the energy theory of localization in metals can be found

in two companion papers (Cherukuri and Shawki,

1995a,b). These papers, pointed out to us during the

reviewing process, show how engineering develop-

ments parallel the recent advances in understanding

ductile shear zones in Earth sciences. It is not surpris-

ing that the basic conclusions are compatible, thus

giving an incentive for future research across the two

disciplines. Cherukuri and Shawki postulate that local-

ization phenomena in thermal-elasto-viscoplastic

materials can be fully assessed by three independent

numbers affecting the energy equation. The first num-

ber describes thermal conduction and is the local Peclet

number, the second number is the mechanical dissipa-

tion or local shear heating number and the third number

the local Reynolds number describing the local level of

kinetic energy achieved during deformation.

The last dimensionless number defines the biggest

difference between ductile Earth- and metal-deforma-

tion processes although similar conditions can be

recovered in ductile earthquakes as shown in this

review. Metals conduct heat very rapidly so that they

have to be deformed under high Reynolds numbers to

be close to adiabatic conditions. Such near-adiabatic

conditions are found to be a necessary ingredient for

flow localization in metals (Rogers, 1979). For Earth-

like parameters, we, however, advise a different three-

dimensional localization space, leaving the Peclet Pe

and dissipation numbers Di as important ingredients

but adding the damage parameter creating new surface

energy instead of the Reynolds number. In the follow-

ing, we will review the effects of shear heating and

conduction only. For this, we rewrite the energy equa-

tion in a non-dimensional formwhere the subscript ‘‘0’’

refers to a reference value of the field quantities, which

is chosen to be a value where a homogenous solution

applies. In order to use this number as a localization

criterion, it is convenient to define a critical dissipation

and Peclet number with reference to a homogeneous

state just before reaching meta-stability:

DT

Dt¼ DirijVeij �

1

Pejr2T ðA35Þ

where we neglect the dissipation due to volume

changes and the dissipation number is

Di ¼ wr0Ve0qcpT0

t0 ðA36Þ

The dissipation number Di is the ratio of thermal

energy produced by shear heating in the time interval

t0 over the energy required to raise the temperature to

T0. In terms of a thermodynamically based criterion

for departures from the elliptical solutions in extended

Gibbs space (Eq. (A21)), we would need to specify

both critical Peclet Pe and dissipation numbers Di. It

turns out that the flow localization phenomenon is

only weakly dependent on the Peclet number (Rege-

nauer-Lieb and Yuen, 2003) so that we suggest to

characterize ductile materials by a critical dissipation

number to describe their tendency to localization. The

Peclet number Pe gives the ration of heat transfer by

advection over the heat transfer by conduction

Pe ¼ v0rT0

jr2T0¼ v0L0

jðA37Þ

where v0 is the relative velocity of the reference

volume (fluid parcel) with respect to a neighbouring

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349342

volume and L0 its length scale. Since the post-bifurca-

tional shear zone width is only weakly dependent on

the dissipation number but chiefly depends on the

Peclet number (Regenauer-Lieb and Yuen, 2003), a

good material description can be given by quoting the

equilibrium Peclet number for a post-bifurcational

steady state solution. The subscript ‘‘0’’ would in this

case refer the post-bifurcational thermodynamic equi-

librium state. Characterizing the behaviour of com-

plex, composite rheology of the lithosphere in terms

of critical dissipation number and equilibrium Peclet

number opens the way to incorporate the localization

behaviour of the complete lithosphere rheology,

obtained from high resolution feedback calculations,

into a generic upscaled rheology useful for large

scale-coupled mantle convection calculations, which

could be a solution to the problems described in the

section on applications to plate tectonics.

In engineering sciences, a simple energy theory for

localization has been developed much earlier. Shawki

(Shawki, 1994a,b) neglects the thermal-elastic feed-

back term and uses Eq. (A35), solved together with

the momentum and the continuity equations, to come

up with an energy theory of localization for a single

fault in pure shear, i.e. the class of 1D shear zone

models discussed in the review and shown schemat-

ically in Fig. 2. He solves by linear stability analysis a

perturbed initial solution of homogenous simple shear

flow. Using the rheologies discussed here, he derives a

critical energy criterion and a critical wavelength

threshold for growth of perturbations, which also

serves as a scaling length for shear zone width.

Shawki’s energy criterion for localization is a

variance to the one proposed in Eq. (A21). Shawki

uses the fact that prior to visible flow localization

stationary elastic body waves are emitted, finally

guiding visco-plastic bifurcation. This has already

been pointed out for the classical constitutive theory

in the elasto-plastic case. A good example showing an

elastic energy wave preceding elasto-visco-plastic

shear zone formation is shown in Fig. 5 of Regena-

uer-Lieb and Yuen (2000b). While this effect is very

difficult to capture numerically, we have suggested to

rephrase the criterion into a thermodynamically based

approach described by a pair of critical dissipation

number and Peclet numbers needed for onset of

localization. When simplifying the analytical results

of Shawki, the dissipation number can be isolated as

the critical number, which is also valid as a global

criterion for many interacting faults with more com-

posite complicated lithosphere rheology (Regenauer-

Lieb and Yuen, 2003). The results of Shawki and

coworkers differs from our afore mentioned papers

only concerning the role of elasticity for shear zone

nucleation. Shawki concludes that elasticity does not

enter the shear zone nucleation criterion but plays an

important role on the subsequent shear zone width.

This is true if the thermal-elastic feedback term is

neglected in the energy equation. We find that with

thermal-elastic feedback all localization phenomena

require three orders of magnitudes lower dissipation

numbers than in comparable cases without thermal-

elasticity. Thermal-elasticity acts like a booster to hete-

rogeneous, thermal–mechanical, ductile shear zones.

Of particular scientific concern is the well-posed-

ness of the scientific problem. For this a proof of

existence of a unique homogenous solution for the

initial boundary conditions must be given. We have

mentioned that, in terms of thermodynamics, a posi-

tive intrinsic dissipation (Eq. (A21)) is a necessary but

not a completely sufficient condition for the existence

of a homogenous solution. For the simple shear 1D

case, Shawki showed that a unique exact homoge-

neous solution exists for velocity controlled bound-

aries only if adiabatic (thermally insulating)

boundaries are selected. We found equivalent homo-

geneous solutions for the case of a similar pure shear

setup (Regenauer-Lieb and Yuen, 2003).

We conclude that Shawki’s energy theory, with the

suggested amendments, is a suitable approach for

geological materials, if we assume only ‘‘simple’’

feedback between conduction and shear heating. Lo-

calization in such simple ductile materials appears to be

entirely controlled by the two non-dimensional numb-

ers appearing in the truncated energy Eq. (A35).

Considering the low values of diffusivity of rocks,

the two numbers have different implications. While

the critical dissipation number for transition from

homogeneous to bifurcating solutions is giving a cri-

terion for onset of localization the Peclet number is

controlling the final width of thermo-mechanical shear

zones. These two quantities describe intrinsic thermo-

dynamic functions and therefore are constitutive prop-

erties, too. But in contrast to the classical brittle theory

the ductile theory of localization relies entirely on

energy fluxes.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 343

References

Abe, S., Mora, P., Place, D., 2000. Extension of the lattice solid

model to incorporate temperature related effects. Pure and Ap-

plied Geophysics 157 (11–12), 1867–1887.

Aifantis, E.C., 1987. The physics of plastic deformation. Interna-

tional Journal of Plasticity 3, 211–247.

Albert, R.A., Phillips, R.J., Dombard, A.J., Brown, C.D., 2000. A

test of the validity of yield strength envelopes with an elasto-

viscoplastic finite element model. Geophysical Journal Interna-

tional 140 (2), 399–409.

Ananthakrishna, G., Bharathi, M.S., Fressengeas, C., Kubin, L.P.,

Lebyodkin, M., 2001. Scale transitions in the dynamic analysis

of jerky flow. Journal de Physique IV 11 (PR5), 135–142.

Anders, M.H., Sleep, N.H., 1992. Magmatism and extension—the

thermal and mechanical effects of the yellowstone hotspot.

Journal of Geophysical Research—Solid Earth 97 (B11),

15379–15393.

Ashby, M.F., Verall, R.A., 1977. Micromechanisms of flow and

fracture, and their relevance to the rheology of the upper mantle.

Philosophical Transactions of the Royal Society of London 288,

59–95.

Backofen, W.A., 1972. Deformation Processing. Addison-Wesley,

Reading. 326 pp.

Balachandar, S., Yuen, D.A., Reuteler, D., Lauer, G., 1995. Viscous

dissipation in three dimensional convection with temperature

dependent viscosity. Science 267, 1150–1153.

Bauer, P., Palm, S., Handy, M.R., 2000. Strain localization and fluid

pathways in mylonite: inferences from in situ deformation of a

water-bearing quartz analogue (norcamphor). Tectonophysics

320 (2), 141–165.

Beaumont, C., Ellis, S., Hamilton, J., Fulsack, P., 1996a. Mechan-

ical model for subduction –collision tectonics of alpine-type

compressional orogens. Geology 24, 657–678.

Beaumont, C., Kamp, P.J.J., Hamilton, J., Fullsack, P., 1996b. The

continental collision zone, South Island, New Zealand: compar-

ison of geodynamical models and observations. Journal of Geo-

physical Research 101 (B2), 3333–3359.

Ben-Zion, Y., Lyakhovski, V., 2002. Accelerated seismic release

and related aspects of seismicity patterns on earthquake faults.

Pure and Applied Geophysics 159, 2385–2412.

Ben-Zion, Y., Sammis, C.G., 2003. Characterization of fault zones.

Pure and Applied Geophysics 160, 677.

Bercovici, D., 1993. A simple model of plate generation from man-

tle flow. Geophysical Journal International 114, 635–650.

Bercovici, D., 1996. Plate generation in a simple model of litho-

sphere–mantle flow with dynamic self-lubrication. Earth and

Planetary Science Letters 144, 41–51.

Bercovici, D., 1998. Generation of plate tectonics from litho-

sphere–mantle flow and void-volatile self-lubrication. Earth

and Planetary Science Letters 154, 139–151.

Bercovici, D., 2002. The generation of plate tectonics from mantle

convection. Earth and Planetary Science Letters 6451, 1–15

(Frontiers).

Bercovici, D., Ricard, Y., 2003. Energetics of a two-phase model of

lithospheric damage, shear localization and plate-boundary for-

mation. Geophysical Journal International 152, 581–596.

Bercovici, D., Ricard, Y., Schubert, G., 2001a. A two-phase model

for compaction and damage: 1. General theory. Journal of Geo-

physical Research-Solid Earth 106 (B5), 8887–8906.

Bercovici, D., Ricard, Y., Schubert, G., 2001b. A two-phase model

for compaction and damage: 3. Applications to shear localiza-

tion and plate boundary formation. Journal of Geophysical Re-

search—Solid Earth 106 (B5), 8925–8939.

Bird, P., 1978. Finite elements modeling of lithosphere deformation:

the Zagros collision orogeny. Tectonophysics 50, 307–336.

Bird, P., 1988. Formation of the Rocky Mountains, western United

States: a continuum computer model. Science 239, 1501–1507.

Bird, P., 1998. Testing hypotheses on plate-driving mechanisms

with global lithosphere models including topography, thermal

structure, and faults. Journal of Geophysical Research—Solid

Earth 103 (B5), 10115–10129.

Blanpied, M.L., Tullis, T.E., Weeks, J.D., 1998. Effects of slip, slip

rate, and shear heating on the friction of granite. Journal of

Geophysical Research—Solid Earth 103 (B1), 489–511.

Bons, P.D., Jessel, M.W., Passchier, C.W., 1993. The analysis of

progressive deformation in rock analogues. Journal of Structural

Geology 15, 403–411.

Branlund, J., Regenauer-Lieb, K., Yuen, D., 2000. Fast ductile fail-

ure of passive margins from sediment loading. Geophysical

Research Letters 27 (13), 1989–1993.

Branlund, J., Regenauer-Lieb, K., Yuen, D., 2001. Weak zone for-

mation for initiating subduction from thermo-mechanical feed-

back of low-temperature plasticity. Earth and Planetary Science

Letters 190, 237–250.

Braun, J., Chery, J., Poliakov, A.N.B., Mainprice, D., Vauchez, A.,

Tommasi, A., Daignieres, M., 1999. A simple parameterization

of strain localization in the ductile regime due to grain size

reduction: a case study for olivine. Journal of Geophysical Re-

search 104 (B11), 25167–25181.

Buck, W.R., Poliakov, A.N.B., 1998. Abyssal hills formed by

stretching oceanic lithosphere. Nature 392, 272–275.

Bulatov, V., Abraham, F.F., Kubin, L., Devincre, B., Yip, S., 1998.

Connecting atomistic and mesoscale simulations of crystal plas-

ticity. Nature 391, 669–672.

Burov, E., Jolivet, L., Le Pourhiet, L., Poliakov, A., 2001. A ther-

momechanical model of exhumation of high pressure (HP) and

ultra-high pressure (UHP) metamorphic rocks in Alpine-type

collision belts. Tectonophysics 342 (1–2), 113–136.

Chakrabarty, J., 2000. Applied Plasticity. Springer, Berlin. 682 pp.

Cherukuri, H.P., Shawki, T.G., 1995a. An energy-based localization

theory: I. Basic framework. International Journal of Plasticity

11, 15–40.

Cherukuri, H.P., Shawki, T.G., 1995b. An energy-based localization

theory: II. Effects of the diffusion, inertia, and dissipation num-

bers. International Journal of Plasticity 11, 41–64.

Chery, J., Vilotte, J.P., Daigniers, M., 1991. Thermomechanical

evolution of a thinned continental lithosphere under compres-

sion: implication for the Pyrenees. Journal of Geophysical Re-

search 96 (B3), 4385–4412.

Chester, F.M., 1995. A rheologic model for wet crust applied to

strike–slip faults. Journal of Geophysical Research 100 (B7),

13033–13044.

Chester, F.M., Higgs, N.G., 1992. Multimechanism friction con-

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349344

stitutive model for ultrafine quartz gouge at hypocentral con-

ditions. Journal of Geophysical Research—Solid Earth 97 (B2),

1859–1870.

Christensen, U.R., 1992. An Eulerian technique for thermal–me-

chanical modeling of lithospheric extension. Journal of Geo-

physical Research 97, 2015–2036.

Christiansen, P.P., Pollard, D.D., 1997. Nucleation, growth and

structural development of mylonitic shear zones in granitic rock.

Journal of Structural Geology 19 (9), 1159–1172.

Christiansen, P.P., Pollard, D.D., 1998. Nucleation, growth and

structural development of mylonitic shear zones in granitic

rocks: reply. Journal of Structural Geology 20 (12), 1801–1803.

Chrysochoos, A., Belmahjoub, F., 1992. Thermographic analysis

of thermomechanical couplings. Archives Mechanics 44 (1),

55–68.

Chrysochoos, A., Peyroux, R., 1997. Modelisation numerique des

couplages en thermomecanique des solides. Revue europeene

des element finis 6 (5–6), 673–724.

Cloetingh, S.A.P.L., Wortel, M.J.R., Vlaar, N.J., 1982. Evolution of

passive continental margins and initiation of subduction zones.

Nature 297, 139–142.

Cloetingh, S., Burov, E., Poliakov, A., 1999. Lithosphere folding:

primary response to compression? (From central Asia to Paris

basin). Tectonics 18 (6), 1064–1083.

Coffin, L.F., Rogers, H.C., 1967. Influence of pressure on the struc-

tural damage in metal forming processes. Transaction of the

American Society of Metals 60, 672–683.

de Borst, R., 2002. Fracture in quasi-brittle materials: a review of

continuum damage-based approaches. Engineering Fracture Me-

chanics 69 (2), 95–112.

DeMets, C., Gordon, R.G., Argus, D.F., Stein, S., 1990. Current

plate motions. Geophysical Journal International 101, 425–478.

Dewhurst, P., Collins, I.F., 1973. A matrix technique for construct-

ing slip-line field solutions to a class of plane strain plasticity

problems. International Journal of Numerical Methods in Engi-

neering 7, 357–378.

Dieterich, J.H., 1979a. Modeling of rock friction: 1. Experimental

results and constitutive equations. Journal of Geophysical Re-

search 84 (NB5), 2161–2168.

Dieterich, J.H., 1979b. Modeling of rock friction: 2. Simulation of

pre-seismic slip. Journal of Geophysical Research 84 (NB5),

2169–2175.

Dieterich, J.H., 1992. Earthquake nucleation on faults with rate-

dependent and state-dependent strength. Tectonophysics 211

(1–4), 115–134.

Dodd, B., Baiy, Y., 1987. Ductile Fracture and Ductility. Academic

Press, London. 309 pp.

Dorogokupets, P.I., 2001. Equation of state and internally consis-

tent thermodynamic functions of minerals. Petrology 9 (6),

534–544.

Downes, H., 1990. Shear zones in the upper mantle—relations

between geochemical enrichment and deformation in mantle

peridotites. Geology 18, 374–377.

Drury, M.R., Vissers, R.L.M., Vanderwal, D., Strating, E.H.H.,

1991. Shear localization in upper mantle peridotites. Pure and

Applied Geophysics 137 (4), 439–460.

Dubuffet, F., Yuen, D.A., Yanagawa, T., 2000. Feedback effects of

variable thermal conductivity on the cold downwellings in high

Rayleigh number convection. Geophysical Research Letters 27

(18), 2981–2984.

Ellis, S., Beaumont, C., Pfiffner, O.A., 1999. Geodynamic models

of crustal-scale episodic tectonic accretion and underplating in

subduction zones. Journal of Geophysical Research 104 (B7),

15169–15190.

England, P., Houseman, G., 1985. The influence of lithospheric

strength heterogeneities on the tectonics of Tibet and surround-

ing regions. Nature 315, 297–301.

England, P., McKenzie, D., 1982. A thin viscous sheet model for

continental deformation. Geophysical Journal of the Royal As-

tronomical Society 70, 295–321.

Evans, B., 1984. The effect of temperature and impurity content on

indentation hardness of quartz. Journal of Geophysical Research

89 (B6), 4213–4222.

Evans, B., Goetze, C., 1979. The temperature variation of the hard-

ness of olivine and its implications for the polycrystalline yield

stress. Journal of Geophysical Research 84, 5505–5524.

Faccenna, C., Davy, P., Brun, J.P., Funiciello, R., Giardini, D.,

Mattei, M., Nalpas, T., 1996. The dynamic of backarc basins:

an experimental approach to the opening of the Tyrrhenian Sea.

Geophysical Journal International 126, 781–795.

Faccenna, C., Funiciello, F., Giardini, D., Lucente, P., 2001. Epi-

sodic back-arc extension during restricted mantle convection in

the Central Mediterranean. Earth and Planetary Science Letters

187 (1–2), 105–116.

Fleck, N.A., Hutchinson, J.W., 2001. A reformulation of strain

gradient plasticity. Journal of the Mechanics and Physics of

Solids 49 (10), 2245–2271.

Fleitout, L., Froidevaux, C., 1980. Thermal and mechanical evolu-

tion of shear zones. Journal of Structural Geology 2 (1–2),

159–164.

Fletcher, R.C., Hallet, B., 1983. Unstable extension of the litho-

sphere: a mechanical model for basin-and-range structure. Jour-

nal of Geophysical Research 88 (B9), 7457–7466.

Fressengeas, C., Molinari, A., 1987. Instability and localization of

plastic-flow in shear at high-strain rates. Journal of the Mechan-

ics and Physics of Solids 35 (2), 185–211.

Gao, H., Huang, Y., Nix, W.D., 1999. Modeling plasticity at the

micrometer scale. Naturwissenschaften 86 (11), 507–515.

Gerbault, M., Poliakov, A.N.B., Daignieres, M., 1998. Prediction of

faulting from the theories of elasticity and plasticity: what are

the limits? Journal of Structural Geology 20 (2–3), 301–320.

Goetze, C., Evans, B., 1979. Stress and temperature in the bending

lithosphere as constrained by experimental rock mechanics.

Geophysical Journal of the Royal Astronomical Society 59,

463–478.

Gomberg, J., Beeler, N., Blanpied, M., 2000. On rate-state and

Coulomb failure models. Journal of Geophysical Research-Solid

Earth 105 (B4), 7857–7871.

Gordon, R.G., Stein, S., 1992. Global tectonics and space geodesy.

Science 256, 333–342.

Govers, R., Wortel, M.J.R., 1995. Extension of stable continental

lithosphere and the initiation of lithospheric scale faults. Tec-

tonics 14 (4), 1041–1055.

Green, A.E., Naghdi, P.M., 1965. A dynamical theory of interacting

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 345

continua. Archives of Rational Mechanics and Analysis 24,

243–263.

Griggs, D.T., Turner, F.J., Heard, H.C., 1960. Deformation of rocks

at 500 jC and 800 jC. Geological Society of America Memoir

79, 39–104.

Grosfils, E.B., Head, J.W., 1996. The timing of giant radiating

dikeswarms emplacement on Venus: implications for resurfac-

ing of the planet and its subsequent evolution. Journal of Geo-

physical Research 101, 4645–4656.

Gruntfest, I.J., 1963. Thermal feedback in liquid flow–plane shear

at constant stress. Transactions of the Society of Rheology 7,

195–207.

Guo, Y., Huang, Y., Gao, H., Zhuang, Z., Hwang, K.C., 2001.

Taylor-based nonlocal theory of plasticity: numerical studies

of the micro-indentation experiments and crack tip fields.

International Journal of Solids and Structures 38 (42–43),

7447–7460.

Gurnis, M., Muller, R.D., Moresi, L., 1998. Cretaceous vertical

motion of Australia and the Australian–Antarctic discordance.

Science 279 (5356), 1499–1504.

Hall, C.E., Parmentier, E.M., 2003. The influence of grain size

evolution on convective instability. G3 (in press).

Handy, M.R., 1994. The energetics of steady state heterogeneous

shear in mylonitic rock. Materials Science and Engineering, A

175, 261–272.

Handy, M.R., Braun, J., Brown, M., Kukowski, N., Paterson, M.S.,

Schmid, S.M., Stockhert, B., Stuwe, K., Thompson, A.B., Wos-

nitza, E., 2001. Rheology and geodynamic modelling: the next

step forward. International Journal of Earth Sciences 90 (1),

149–156.

Hill, R., 1950. The Mathematical Theory of Plasticity. Oxford Univ.

Press, London. 356 pp.

Hirsch, P.B., 1975. Work hardening. In: Hirsch, P.B. (Ed.),

The Physics of Metals. Cambridge Univ. Press, Cambridge,

pp. 189–246.

Hobbs, B.E., Ord, A., 1988. Plastic instabilities: implications for the

origin of intermediate and deep focus earthquakes. Journal of

Geophysical Research 93 (B9), 10521–10540.

Hobbs, B.E., Ord, A., Teyssier, C., 1986. Earthquakes in the ductile

regime. Pure and Applied Geophysics 124 (1/2), 310–336.

Hochstein, M.P., Regenauer-Lieb, K., 1998. Heat generation asso-

ciated with the collision of two plates: the Himalaya Geother-

mal Belt. Journal of Volcanology and Geothermal Research 83

(1–2), 75–92.

Hochstein, M.P., Smith, I.E.M., Regenauer-Lieb, K., Ehara, S.,

1993. Geochemistry and heat transfer processes in Quaternary

rhyolitic systems of the Taupo Volcanic Zone, New Zealand.

Tectonophysics 223, 213–235.

Holian, B., Lomdahl, P., 1998. Plasticity induced by shock waves in

non-equilibrium molecular-dynamics simulations. Science 280,

2085–2088.

Huismans, R.S., Beaumont, C., 2002. Asymmetric lithospheric ex-

tension: the role of frictional plastic strain softening inferred

from numerical experiments. Geology 30 (3), 211–214.

Jimenez-Munt, I., Bird, P., Fernandez, M., 2001. Thin-shell model-

ing of neotectonics in the Azores–Gibraltar region. Geophysical

Research Letters 28 (6), 1083–1086.

Jin, D.H., Karato, S., Obata, M., 1998. Mechanisms of shear local-

ization in the continental lithosphere: inference from the defor-

mation microstructures of peridotites from the Ivrea zone,

northwestern Italy. Journal of Structural Geology 20 (2–3),

195–209.

Johnson, W., Baraya, G.L., Slater, R.A.C., 1964. On heat lines or

lines of thermal discontinuity. International Journal of Mechan-

ical Sciences 6, 409–507.

Johnson, W., Sowerby, R., Venter, R.D., 1982. Plane Strain Slip

Line Fields for Metal Deformation Processes. Pergamon, Ox-

ford. 364 pp.

Jung, H., Karato, S.I., 2001. Effects of water on dynamically re-

crystallized grain-size of olivine. Journal of Structural Geology

23 (9), 1337–1344.

Kameyama, C., Kaneda, Y., 2002. Thermal–mechanical coupling

in shear deformation as a model of frictional constitutive rela-

tions. Pure and Applied Geophysics 159 (9), 2011–2028.

Kameyama, M.C., Yuen, D.A., Fujimoto, H., 1997. The interaction

of viscous heating with grain-size dependent rheology in the

formation of localized slip zones. Geophysical Research Letters

24 (20), 2523–2526.

Kameyama, C., Yuen, D.A., Karato, S., 1999. Thermal–mechanical

effects of low temperature plasticity (the Peierls mechanism) on

the deformation of a viscoelastic shear zone. Earth and Planetary

Science Letters 168, 159–162.

Kanamori, H., Anderson, D.L., Heaton, T.H., 1998. Frictional melt-

ing during the rupture of the 1994 Bolivian earthquake. Science

279 (5352), 839–842.

Karato, S.I., 1989. Grain growth kinetics in olivine aggregates.

Tectonophysics 168, 253–257.

Karato, S., Riedel, M.R., Yuen, D.A., 2001. Rheological structure

and deformation of subducted slabs in the mantle transition

zone: implications for mantle circulation and deep earthquakes.

Physics of the Earth and Planetary Interiors 127 (1–4), 83–108.

Kato, N., 2001. Effect of frictional heating on pre-seismic sliding: a

numerical simulation using a rate-, state- and temperature-de-

pendent friction law. Geophysical Journal International 147 (1),

183–188.

Kohlstedt, D.L., Mackwell, S.J., 1998. Diffusion of hydrogen and

intrinsic point defects in olivine. Zeitschrift Fur Physikalische

Chemie—International Journal of Research in Physical Chem-

istry and Chemical Physics 207, 147–162.

Kohlstedt, D.L., Evans, B., Mackwell, S.J., 1995. Strength of the

lithosphere: constraints imposed by laboratory measurements.

Journal of Geophysical Research 100 (B9), 17587–17602.

Kusznir, N.J., 1982. Lithosphere response to externally and inter-

nally derived stresses—a viscoelastic stress guide with amplifi-

cation. Geophysical Journal of the Royal Astronomical Society

70 (2), 399–414.

Lavenda, B.H., 1978. Thermodynamics of Irreversible Processes.

Macmillan, London. 182 pp.

Lebyodkin, M., Fressengeas, C., Ananthakrishna, G., Kubin, L.P.,

2001. Statistical and multifractal analysis of the Portevin-Le

Chatelier effect. Materials Science and Engineering a-Structur-

al Materials Properties Microstructure and Processing 319,

170–175.

Lemonds, J., Needleman, A., 1986. Finite element analyses of shear

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349346

localization in rate and temperature dependent solids. Mechanics

of Materials 5, 339–361.

Lenardic, A., Kaula, W.M., Bindschadler, D.L., 1995. Some effects

of a dry crustal flow law on numerical simulations of coupled

crustal deformation and mantle convection on Venus. Journal of

Geophysical Research-Planets 100 (E8), 16949–16957.

Lenardic, A., Moresi, L., Muhlhaus, H., 2000. The role of mobile

belts for the longevity of deep cratonic lithosphere: the crumple

zone model. Geophysical Research Letters 27 (8), 1235–1238.

Li, S.F., Liu, W.K., 2000. Numerical simulations of strain local-

ization in inelastic solids using mesh-free methods. Interna-

tional Journal for Numerical Methods in Engineering 48 (9),

1285–1309.

Locket, J.M., Kusznir, N.J., 1982. Ductile shear zones: some as-

pects of constant slip velocity and constant shear stress models.

Geophysical Journal of the Royal Astronomical Society 69,

477–494.

Ludwik, P., 1909. Elemente der Technologischen Mechanik.

Springer Verlag, Berlin.

Luyendyk, B.P., Hornaflus, J.S., 1987. Neogene crustal rotation,

fault slip and basin development in southern California. In: In-

gersoll, R.V., Ernst, R.W.G. (Eds.), Cenozoic Basin Develop-

ment of Coastal California. Prentice Hall, Englewood Cliffs, NJ,

pp. 259–283.

Luyendyk, B.P., Kamerling, M.J., Terres, R.R., Hornaflus, J.S.,

1985. Simple shear of southern California during Neogene time

suggested by paleomagnetic declinations. Journal of Geophys-

ical Research 90 (B12), 12454–12466.

Lyakhovsky, V., 1997. Non-linear elastic behaviour of damaged

rocks. Geological Journal International 130, 157.

Lyakhovsky, V., Podladchikov, Y., Poliakov, A., 1993. A rheological

model of a fractured solid. Tectonophysics 226 (1–4), 187–198.

Lyakhovsky, V., Ben-Zion, Y., Agnon, A., 1997. Distributed dam-

age, faulting, and friction. Journal of Geophysical Research—

Solid Earth 102 (B12), 27635–27649.

Lyakhovsky, V., Ben-Zion, Y., Agnon, A., 2001. Earthquake cycle,

fault zones, and seismicity patterns in a rheologically layered

lithosphere. Journal of Geophysical Research-Solid Earth 106

(B3), 4103–4120.

Lyzenga, G.A., Wallace, K.S., Fanselo, J.L., Raefsky, A., Groth,

P.M., 1986. Tectonic motions in California inferred from very

long baseline interferometry observations. Journal of Geophys-

ical Research 91, 9473–9487.

Mandl, G., Jong, L.N.J., Maltha, A., 1977. Shear zones in granular

material. Rock Mechanics. Springer, Berlin, pp. 95–144.

Marotta, A.M., Bayer, U., Scheck, M., Thybo, H., 2001. The stress

field below the NE German Basin: effects induced by the Alpine

collision. Geophysical Journal International 144 (2), F8–F12.

Matsumoto, N., Yomogida, K., Honda, S., 1992. Fractal analysis of

fault systems in Japan and the Philippines. Geophysical Re-

search Letters 19 (4), 357–360.

McKenzie, D.P., Brune, J.N., 1972. Melting on fault planes during

large earthquakes. Geophysical Journal of the Royal Astronom-

ical Society 29, 65–78.

Mei, S., Kohlstedt, D.L., 2000. Influence of water on plastic defor-

mation of olivine aggregates: 1. Diffusion creep regime. Journal

of Geophysical Research 105 (B9), 21457–21469.

Melosh, H.J., 1976. Plate motion and thermal-instability in astheno-

sphere. Tectonophysics 35 (4), 363–390.

Monaghan, J.J., 1992. Smoothed particle hydrodynamics. Astron-

omy and Astrophysics 30, 543–574.

Montesi, L.G.J., Hirth, G., 2003. Transient behavior of a ductile

shear zone: from laboratory experiments to postseismic creep.

Earth and Planetary Science Letters (submitted for publication).

Montesi, L.G.J., Zuber, M.T., 2002. A unified description of local-

ization for application to large-scale tectonics. Journal of Geo-

physical Research—Solid Earth 107 (B3), 2045.

Mora, P., Place, D., 1998. Numerical simulation of earthquake

faults with gouge: toward a comprehensive explanation for the

heat flow paradox. Journal of Geophysical Research-Solid Earth

103 (B9), 21067–21089.

Moresi, L., Solomatov, V., 1998. Mantle convection with a brittle

lithosphere: thoughts on the global tectonic styles of the Earth

and Venus. Geophysical Journal International 133 (3), 669–682.

Munjiza, A., John, N.W.M., 2002. Mesh size sensitivity of the

combined FEM/DEM fracture and fragmentation algorithms.

Engineering Fracture Mechanics 69 (2), 281–295.

Nakazaki, Y., Tanaka, Y., Goto, N., Inui, T., 1995. Mechanisms of

CO2 separation by microporous crystals estimated by computa-

tional chemistry. Catalysis Today 23 (4), 391–396.

Needleman, A., 1994. Computational modeling of material failure.

Applied Mechanics Reviews 47, 34–42.

Needleman, A., Tvergaard, V., 1992. Analyses of plastic flow de-

formation in metals. Applied Mechanics Reviews 45, 3–18.

Nye, J.F., 1953. The flow law of ice from measurements in glacier

tunnels, laboratory experiments and the jungfraufirn borehole

experiment. Proceedings of the Royal Society of London Series

A, Mathematical and Physical Sciences 219 (1139), 477–489.

Ogawa, M., 1987. Shear instability in a viscoelastic material as the

cause of deep focus earthquakes. Journal of Geophysical Re-

search 92 (B1), 13801–13810.

Okubo, P.G., Aki, K., 1987. Fractal geometry in the San-Andreas

fault system. Journal of Geophysical Research—Solid Earth and

Planets 92 (B1), 345–355.

Orowan, E., 1960. Mechanism of seismic faulting. In: Griggs, D.,

Handin, J. (Eds.), Rock Deformation (A Symposium), pp. 323–

346. Washington.

Petit, J.P., Wibberley, C.A.J., Ruiz, G., 1999. ‘Crack-seal’, slip: a

new fault valve mechanism? Journal of Structural Geology 21

(8–9), 1199–1207.

Petrini, K., Podladchikov, Y., 2000. Lithospheric pressure–depth

relationship in compressive regions of thickened crust. Journal

of Metamorphic Geology 18 (1), 67–77.

Pitzer, K.S., Sterner, S.M., 1994. Equations of state valid continu-

ously from zero to extreme pressures for H2O and CO2. Journal

of Chemistry and Physics 101, 3111–3116.

Poirier, J.P., Tarantola, A., 1998. A logarithmic equation of state.

Physics of the Earth and Planetary Interiors 109 (1–2), 1–8.

Poliakov, A.N.B., Herrmann, H.J., 1994. Self-organized criticality

of plastic shear bands in rocks. Geophysical Research Letters 21

(19), 2143–2146.

Poliakov, A.N.B., Herrmann, H.J., Podladchikov, Y.Y., 1994. Frac-

tal plastic shear bands. Fractals 2, 567–581.

Pollitz, F.F., 2001. Viscoelastic shear zone model of a strike–slip

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 347

earthquake cycle. Journal of Geophysical Research-Solid Earth

106 (B11), 26541–26560.

Post, R.L., 1977. High temperature creep of Mt. Burnet dunite.

Tectonophysics 42, 75–110.

Povirk, G.L., Nutt, S.R., Needleman, A., 1994. Continuum model-

ling of residual stresses in metal-matrix composites. In: Barrera,

B.V., Dutta, I. (Eds.), Residual Stresses in Composites. TMS,

New York, pp. 3–23.

Ranalli, G., 1995. Rheology of the Earth. Chapman and Hall, Lon-

don. 413 pp.

Regenauer-Lieb, K., 1999. Dilatant plasticity applied to Alpine

Collision: ductile void growth in the intraplate area beneath

the Eifel Volcanic Field. Journal of Geodynamics 27, 1–21.

Regenauer-Lieb, K., Kohl, T., 2003. Water solubility and diffusivity

in olivine: its role for planetary tectonics. Mineralogical Maga-

zine.

Regenauer-Lieb, K., Petit, J.P., 1997. Cutting of the European

continental lithosphere: plasticity theory applied to the pre-

sent Alpine collision. Journal of Geophysical Research 102,

7731–7746.

Regenauer-Lieb, K., Yuen, D., 1998. Rapid conversion of elastic

energy into shear heating during incipient necking of the litho-

sphere. Geophysical Research Letters 25 (14), 2737–2740.

Regenauer-Lieb, K., Yuen, D., 2000a. Quasi-adiabatic instabilities

associated with necking processes of an elasto-viscoplastic litho-

sphere. Physics of the Earth and Planetary Interiors 118, 89–102.

Regenauer-Lieb, K., Yuen, D.A., 2000b. Fast mechanisms for the

formation of new plate boundaries. Tectonophysics 322, 53–67.

Regenauer-Lieb, K., Yuen, D.A., 2003. Positive feedback of inter-

acting ductile faults from coupling of equation of state, rheology

and thermal-mechanics. Physics of Earth and Planetary Interi-

ors, submitted.

Regenauer-Lieb, K., Yuen, D., Branlund, J., 2001. The initation

of subduction: criticality by addition of water? Science 294,

578–580.

Ricard, Y., Froidevaux, C., 1986. Stretching instabilities and litho-

spheric boudinage. Journal of Geophysical Research 91 (B8),

8314–8324.

Ricard, Y., Bercovici, D., Schubert, G., 2001. A two-phase model

for compaction and damage: 2. Applications to compaction,

deformation, and the role of interfacial surface tension. Journal

of Geophysical Research—Solid Earth 106 (B5), 8907–8924.

Rice, J.R., 1977. The localization of plastic deformation. In: Koiter,

W.T. (Ed.), Theoretical and Applied Mechanics. North-Holland,

Amsterdam, pp. 207–220.

Roberts, D.C., Turcotte, D.L., 2000. Earthquakes: friction or a

plastic instability? In: Rundle, J.B., Turcotte, D.L., Klein, W.

(Eds.), GeoComplexity and the Physics of Earthquakes. Amer-

ican Geophysical Union, Washington, DC, pp. 97–104.

Roedder, E., 1981. Origin of fluid inclusions and changes that occur

after trapping. In: Hollister, L.S., Crawford, M.L. (Eds.), Short

Course in Fluid Inclusions: Applications to Petrology. Minera-

logical Society of Canada, Calgary, pp. 101–137.

Roeder, E., 1965. Fluid inclusions. American Mineralogist 50,

1746–1782.

Rogers, H.C., 1979. Adiabatic plastic deformation. Annual Review

of Material Science 9, 283–311.

Rudnicki, J.W., Rice, J.R., 1975. Conditions for localization of

deformation in pressure-sensitive dilatant materials. Journal of

the Mechanics and Physics of Solids 23 (6), 371–394.

Rundle, P.B., Rundle, J.B., Tiampo, K.F., Martins, J.S.S., McGin-

nis, S., Klein, W., 2001. Nonlinear network dynamics on earth-

quake fault systems—art. no. 148501. Physical Review Letters

8714 (14), 148501/1–148501/4.

Scholz, C.H., 2000a. Evidence for a strong San Andreas fault.

Geology 28 (2), 163–166.

Scholz, C.H., 2000b. A fault in the ‘weak San Andreas’ theory.

Nature 406 (6793), 234.

Schott, B., Yuen, D.A., Schmeling, H., 2000. The significance of

shear heating in continental delamination. Physics of the Earth

and Planetary Interiors 118 (3–4), 273–290.

Schubert, G., Yuen, D.A., 1977. Possibility of thermal-instability in

shear flows in Earths Upper Mantle. Bulletin of the American

Physical Society 22 (10), 1281.

Schubert, G., Yuen, D.A., 1978. Shear heating instability in Earths

Upper Mantle. Tectonophysics 50 (2–3), 197–205.

Schubert, G., Froidevaux, C., Yuen, D.A., 1976. Oceanic litho-

sphere and astenosphere: thermal and mechanical structure.

Journal of Geophysical Research 81 (20), 3525–3540.

Shawki, T.G., 1994a. An energy criterion for the onset of shear

localization in thermal viscoplastic material: Part I. Necessary

and sufficient initiation conditions. Journal of Applied Mechan-

ics 61, 530–537.

Shawki, T.G., 1994b. An energy criterion for the onset of shear

localization in thermal viscoplastic material: Part II. Applications

and implications. Journal of Applied Mechanics 61, 538–547.

Shemenda, A.I., Grocholsky, A.L., 1992. Physical modelling of

lithosphere subduction in collision zones. Tectonophysics 216,

273–290.

Shemenda, A.I., Grocholsky, A.L., 1994. Physical modeling of

slow seafloor spreading. Journal of Geophysical Research 99

(B5), 9137–9153.

Sherif, R.A., Shawki, T.G., 1992. The role of heat conduction dur-

ing the post-localization regime in dynamic viscoplasticity. In:

Zbib, H.M. (Ed.), Plasticity and Creep. ASME, New York,

pp. 159–173.

Shimada, M., 1993. Lithosphere strength inferred from fracture

strength of rocks at high confining pressures and temperatures.

Tectonophysics 217, 55–64.

Sieh, K., Jones, L., Hauksson, E., Hudnut, K., Eberhartphillips, D.,

Heaton, T., Hough, S., Hutton, K., Kanamori, H., Lilje, A.,

Lindvall, S., McGill, S.F., Mori, J., Rubin, C., Spotila, J.A.,

Stock, J., Thio, H.K., Treiman, J., Wernicke, B., Zachariasen,

J., 1993. Near-field investigations of the landers earthquake

sequence, April to July 1992. Science 260 (5105), 171–176.

Solomatov, V.S., 2001. Grain size-dependent viscosity convection

and the thermal evolution of the Earth. Earth and Planetary

Science Letters 191 (3–4), 203–212.

Spohn, T., 1980. Orogenic volcanism caused by thermal runaways?

Geophysical Journal of the Royal Astronomical Society 62,

403–419.

Tackley, P., 1998. Self-consistent generation of tectonic plates in

three-dimensional mantle convection. Earth and Planetary Sci-

ence Letters 157, 9–22.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349348

Tackley, P., 2000a. Self-consistent generation of tectonic plates in

time- dependent, three-dimensional mantle convection simula-

tions: 1. Pseudoplastic yielding. G3 01 (23), 1525.

Tackley, P., 2000b. Self-consistent generation of tectonic plates in

time-dependent, three-dimensional mantle convection simula-

tions: 2. Strain weakening and asthenosphere. G3 01 (25), 2027.

Tanimoto, T., Okamoto, T., 2000. Change of crustal potential energy

by earthquakes: an indicator for extensional and compressional

tectonics. Geophysical Research Letters 27 (15), 2313–2316.

Tapponnier, P., Molnar, P., 1976. Slip line field theory and large

scale continental tectonics. Nature 264, 319–324.

Tapponnier, P., Molnar, P., 1977. Active faulting and tectonics in

China. Journal of Geophysical Research 82, 2905–2930.

Tikoff, B., Teyssier, C., De Saint Blanquat, M., 1998. Nucleation,

growth and structural development of mylonitic shear zones in

granitic rocks: discussion. Journal of Structural Geology 20

(12), 1795–1799.

Tommasi, A., Vauchez, A., 2001. Continental rifting parallel to

ancient collisional belts: an effect of the mechanical anisotropy

of the lithospheric mantle. Earth and Planetary Science Letters

185 (1–2), 199–210.

Tommasi, A., Vauchez, A., Daudr, B., 1995. Initiation and propa-

gation of shear zones in a heterogeneous continental lithosphere.

Journal of Geophysical Research 100 (B11), 22083–22101.

Trompert, R., Hansen, U., 1998. Mantle convection simulations

with rheologies that generate plate-like behaviour. Nature 395,

686–689.

Tullis, T.E., 1999. Earthquake geophysics—deep slip rates on the

San Andreas fault. Science 285 (5428), 671–672.

Tvergaard, V., 1987. Effect of yield surface curvature and void

nucleation on plastic flow localization. Journal of Mechanics

and Physics of Solids 35, 43–60.

van Daalen, M., Heilbronner, R., Kunze, K., 1999. Orientation

analysis of localized shear deformation in quartz fibres at the

brittle-ductile transition. Tectonophysics 303 (1–4), 83–107.

van Hunen, J., van den Berg, A.P., Vlaar, N.J., 2000. A thermo-

mechanical model of horizontal subduction below an overriding

plate. Earth and Planetary Science Letters 182 (2), 157–169.

Van Swygenhoven, H., 2002. Polycrystalline materials—grain

boundaries and dislocations. Science 296 (5565), 66–67.

Vasilyev, O.V., Podladchikov, Y.Y., Yuen, D.A., 2001. Modelling of

viscoelastic plume– lithosphere interaction using the adaptive

multilevel wavelet collocation method. Geophysical Journal In-

ternational 147 (3), 579–589.

Vermeer, P., 1984. Non-associated plasticity for soils. Concrete and

Rock 29, 1.

Vilotte, J.P., Daignieres, M., Madariaga, R., 1982. Numerical mod-

elling of intraplate deformation: simple mechanical models of

continental collision. Journal of Geophysical Research 87 (B13),

10709–10728.

Vilotte, J.P., Madariaga, R., Daignieres, M., Zienkiewicz, O.C.,

1986. Numerical study of continental collision: influence of

buoyancy forces and a stiff inclusion. Geophysical Journal of

the Royal Astronomical Society 84, 279–310.

Wellman, H.W., 1984. The Alpine Fault, New Zealand, near Milford

Sound and to the southwest. Geological Magazine 5, 437–441.

Wibberley, C.A.J., Petit, J.P., Rives, T., 2000. Mechanics of cata-

clastic ‘deformation band’ faulting in high-porosity sandstone,

Provence. Comptes Rendus De L Academie Des Sciences Serie

Ii Fascicule A, Sciences De La Terre Et Des Planetes 331 (6),

419–425.

Wiens, D.A., Snider, N.O., 2001. Repeating deep earthquakes: evi-

dence for fault reactivation at great depth. Science 293 (5534),

1463–1466.

Williams, C.A., Richardson, R.M., 1991. A rheologically layered

three-dimensional model of the San Andreas fault in Central and

Southern California. Journal of Geophysical Research 96 (B10),

16597–16623.

Wyss, M., Wiemer, S., 2000. Change in the probability for earth-

quakes in southern California due to the Landers magnitude 7.3

earthquake. Science 290 (5495), 1334–1338.

Yuen, D., 2000. Chapter 13: Modeling mantle convection: a sig-

nificant challenge in geophysical fluid dynamics. In: Fox, P.A.,

Kerr, R.M. (Eds.), Geophysical and Astrophysical Convection,

pp. 257–293.

Yuen, D.A., Schubert, G., 1977. Asthenospheric shear flow: ther-

mally stable or unstable? Geophysical Research Letters 4 (11),

503–506.

Yuen, D.A., Fleitout, L., Schubert, G., Froidevaux, C., 1978. Shear

deformation zones along major transform faults and subducting

slabs. Geophysical Journal of the Royal Astronomical Society

54 (1), 93–119.

Yuen, D.A., Vincent, A.P., Bergeron, S.Y., Dubuffet, F., Ten, A.A.,

Steinbach, V.C., Starin, L., 2000. Crossing of scales and non-

linearities in geophysical processes. In: Boschi, E., Ekstrom, G.,

Morelli, A. (Eds.), Problems in Geophysics for the New Millen-

ium. Editrice Compositori, Bologna, Italy, pp. 403–462.

Zbib, H.M., 1992. Plastic Flow and Creep. ASME, New York.

223 pp.

Zhao, D.P., Kanamori, H., 1993. The 1992 landers earthquake se-

quence—earthquake occurrence and structural heterogeneities.

Geophysical Research Letters 20 (11), 1083–1086.

Zhong, S., Gurnis, M., Moresi, L., 1998. Role of faults, nonlinear

rheology, and viscosity structure in generating plates from in-

stantaneous mantle flow models. Journal of Geophysical Re-

search 103 (B7), 15255–15268.

Klaus Regenauer-Lieb is a Research Asso-

ciate/Lecturer at the Institute of Geophysics

ETH Zurich and Privatdozent at the De-

partment of Geosciences of the University

of Mainz. He graduated in Geophysics at

the University Kiel and obtained a PhD in

Geology at the University of Auckland. He

held a post-doctoral in the GEOMAR Kiel,

in the University of Auckland and at the

University of Mainz where he finished in

1999 with a habilitation on energy esti-

mates for large-scale continental deformation. His research interests

are in computational geodynamics, with special interest in the

coupling of lithosphere and mantle dynamics.

K. Regenauer-Lieb, D.A. Yuen / Earth-Science Reviews 63 (2003) 295–349 349

Dave Yuen started out in Physical Chemis-

try and received his Bachelor’s degree from

Caltech. He then switched over to the earth

sciences in the aftermath of plate tectonics

and received a Master’s degree from

Scripps Institution of Oceanography in

1973 and his Doctoral degree under Jerry

Schubert at UCLA in 1978. After spending

2 years with Dick Peltier at the Univ. of

Toronto, Dave went on to Arizona State

University in 1980, then to Univ. Colorado

in 1985, and since 1985 he has been at the University of Minnesota

at both the Minnesota Supercomputing Institute and the Dept. of

Geology and Geophysics. He works on geophysical fluid dynamical

problems ranging from the microscales, using molecular dynamics

to large-scale circulation problems in mantle convection.